Question Bank

1427 questions · 0 solved on this device · all graded live by the simulator.

1427 shown

• Multiple choicecorenew
  A binary source emits symbols with probabilities $1/4$ and $3/4$. What is its Shannon entropy $H = -\sum p \log_2(p)$, in bits?
  advanced · physics · Classical Information and Shannon Entropy
• Multiple choicecorenew
  A single-qubit density matrix is $\rho = \begin{bmatrix}0.5 & 0.25 \\ 0.25 & 0.5\end{bmatrix}$ (real symmetric). What is its von Neumann entropy $S = -\sum \lambda \log_2(\lambda)$, in bits?
  advanced · physics · Von Neumann Entropy
• Multiple choicecorenew
  A 2-qubit system is in the maximally mixed state $\rho = I/4$, i.e. all four eigenvalues equal $1/4$. What is its von Neumann entropy $S = -\sum \lambda \log_2(\lambda)$, in bits?
  advanced · physics · Entropy of Pure and Mixed States
• Multiple choicecorenew
  The joint distribution of (X, Y) is P(0,0) = 1/2, P(0,1) = 1/4, P(1,0) = 0, P(1,1) = 1/4. Using H(Y|X) = H(X,Y) - H(X), what is the conditional entropy H(Y|X) in bits?
  advanced · physics · Joint and Conditional Entropy
• Multiple choicecorenew
  Two qubits are in the maximally entangled Bell state $|\Phi^+\rangle = (|00\rangle + |11\rangle)/\sqrt{2}$. Using $I(A:B) = S(A) + S(B) - S(AB)$, what is the quantum mutual information in bits?
  advanced · physics · Quantum Mutual Information
• True / Falsecorenew
  Subadditivity of the von Neumann entropy states that for any bipartite state rho_AB, S(AB) <= S(A) + S(B), with equality if and only if rho_AB is a product state rho_A (x) rho_B.
  advanced · physics · Subadditivity
• Multiple choicecorenew
  Two qubits are in the pure state $a|00\rangle + b|11\rangle$ with $a = \sqrt{3}/2$ and $b = 1/2$. The reduced state of qubit A is diagonal with eigenvalues $|a|^2$ and $|b|^2$. What is the entanglement entropy $S = -\sum \lambda \log_2(\lambda)$, in bits?
  advanced · physics · Entanglement Entropy
• Multiple choicecorenew
  A two-qubit state has (unnormalized) amplitudes $|00\rangle + |01\rangle + |10\rangle$, each scaled by $1/\sqrt{3}$. Forming the 2x2 coefficient matrix $M$ (rows = A, cols = B), the squared Schmidt coefficients are the eigenvalues of $G = M M^T$, namely about $0.8727$ and $0.1273$. What is the entanglement entropy $S = -\sum \lambda \log_2(\lambda)$, in bits?
  advanced · physics · The Schmidt Decomposition
• Multiple choicecorenew
  The Holevo bound states that for every measurement Bob can make on a quantum ensemble {p_x, rho_x}, the accessible information I(X:Y) is bounded by the Holevo quantity chi. What is chi?
  advanced · physics · Holevo Bound (Overview)
• Multiple choicecorenew
  By Schumacher's quantum source coding theorem, a memoryless quantum source with density operator $\rho$ can be faithfully compressed (fidelity approaching 1) to $R$ qubits per signal exactly when:
  advanced · physics · Data Compression (Schumacher)
• Multiple choicecorenew
  A bipartite pure state has Schmidt coefficients whose squares are the eigenvalues of the reduced density matrix $\rho_A$: $1/3$ and $2/3$. What is the entanglement entropy $S(\rho_A) = -\sum \lambda \log_2(\lambda)$, in bits?
  advanced · physics · Checkpoint: Compute Entropies
• Multiple choicecorenew
  A quantum channel is defined as the most general physically allowed transformation of a quantum state. Which set of three properties exactly characterizes it (a CPTP map)?
  advanced · physics · What Is a Quantum Channel
• True / Falsecorenew
  The transpose map $T(\rho) = \rho^T$ is positive (it preserves the positive semidefiniteness of any state in isolation) but is NOT completely positive, which is why it is not a physical quantum channel.
  advanced · physics · Completely Positive Maps
• Multiple choicecorenew
  A bit-flip channel $E(\rho) = (1-p)\rho + p X \rho X$ with $p = 0.25$ acts on $\rho = |+\rangle\langle+| = \begin{bmatrix}0.5 & 0.5 \\ 0.5 & 0.5\end{bmatrix}$, where $X$ is the Pauli-X gate. What is $\langle 0|E(\rho)|0\rangle$?
  advanced · physics · The Kraus Representation
• Multiple choicecorenew
  The amplitude-damping channel with $g = 0.3$ has Kraus operators $K_0 = \begin{bmatrix}1 & 0 \\ 0 & \sqrt{1-g}\end{bmatrix}$ and $K_1 = \begin{bmatrix}0 & \sqrt{g} \\ 0 & 0\end{bmatrix}$. Compute the completeness sum $M = K_0^\dagger K_0 + K_1^\dagger K_1$ and report the Frobenius distance to the identity, $\sqrt{\sum_{ij} (M_{ij} - I_{ij})^2}$. What is this value?
  advanced · physics · Trace Preservation
• Multiple choicecorenew
  A depolarizing channel $E(\rho) = (1-p) \rho + p (I/2)$ with $p = 0.4$ acts on $\rho = |0\rangle\langle 0|$. What is the purity $\mathrm{tr}(E(\rho)^2)$ of the output?
  advanced · physics · The Depolarizing Channel
• Multiple choicecorenew
  An amplitude-damping channel with damping probability $g = 0.3$ has Kraus operators $K_0 = \begin{bmatrix}1 & 0 \\ 0 & \sqrt{1-g}\end{bmatrix}$ and $K_1 = \begin{bmatrix}0 & \sqrt{g} \\ 0 & 0\end{bmatrix}$. It acts on $\rho = |1\rangle\langle 1|$. What is the resulting excited-state population $\langle 1|E(\rho)|1\rangle$?
  advanced · physics · Amplitude-Damping Channel
• Multiple choicecorenew
  A phase-damping channel with $L = 0.5$ has Kraus operators $K_0 = \begin{bmatrix}1 & 0 \\ 0 & \sqrt{1-L}\end{bmatrix}$ and $K_1 = \begin{bmatrix}0 & 0 \\ 0 & \sqrt{L}\end{bmatrix}$. It acts on $\rho = |+\rangle\langle +| = \begin{bmatrix}0.5 & 0.5 \\ 0.5 & 0.5\end{bmatrix}$. What is the survival probability $\langle +|E(\rho)|+\rangle$?
  advanced · physics · Phase-Damping Channel
• Multiple choicecorenew
  Under the Choi-Jamiolkowski isomorphism, which property of a linear map E corresponds to its Choi operator J(E) being positive semidefinite (J >= 0)?
  advanced · physics · The Choi–Jamiołkowski Isomorphism
• Multiple choicecorenew
  Two amplitude-damping channels are applied in sequence to $\rho = |1\rangle\langle 1|$: first with $g_1 = 0.2$, then with $g_2 = 0.5$. What is the surviving excited-state population $\langle 1|E(\rho)|1\rangle$ after both?
  advanced · physics · Channel Composition
• Multiple choicecorenew
  Stinespring's dilation theorem writes any CPTP channel $E$ as $E(\rho) = \mathrm{tr}_E(V \rho V^\dagger)$ for a linear map $V$ into the system-plus-environment space. What condition must $V$ satisfy, and how large can the environment be taken?
  advanced · physics · The Stinespring Dilation
• Multiple choicecorenew
  A depolarizing channel $E(\rho) = (1-p) \rho + p (I/2)$ with $p = 0.36$ acts on $\rho = |0\rangle\langle 0|$. The fidelity with the input is $F = \sqrt{\langle 0|E(\rho)|0\rangle}$. What is $F$?
  advanced · physics · Channel Fidelity
• Multiple choicecorenew
  An amplitude-damping channel with $g = 0.5$ acts on $\rho = |+\rangle\langle +| = \begin{bmatrix}0.5 & 0.5 \\ 0.5 & 0.5\end{bmatrix}$. What is the purity $\mathrm{tr}(E(\rho)^2)$ of the output state?
  advanced · physics · Checkpoint: Analyze a Channel
• Multiple choicecorenew
  The classically-correlated separable state (1/2)|00><00| + (1/2)|11><11| has reduced-state entropy S(rho_A) = 1 bit. Why is S(rho_A) nevertheless NOT a valid entanglement measure for mixed states?
  advanced · physics · Quantifying Entanglement
• Multiple choicecorenew
  The entanglement of formation $E_F(\rho)$ of a mixed state is defined as the convex roof of the pure-state entanglement. Concretely, over all pure-state ensembles $\{p_i, |\psi_i\rangle\}$ that realize $\rho$, it is:
  advanced · physics · Entanglement of Formation
• Multiple choicecorenew
  For the pure two-qubit state $|\psi\rangle = \cos(\theta)|00\rangle + \sin(\theta)|11\rangle$ with real amplitudes and $\theta = \pi/5$, the concurrence is $C = 2|ad - bc|$ where $(a,b,c,d)$ are the amplitudes of $(|00\rangle,|01\rangle,|10\rangle,|11\rangle)$, reducing to $C = |\sin(2\theta)|$. What is $C$?
  advanced · physics · Concurrence
• Multiple choicecorenew
  A two-qubit Werner state with parameter $p = 2/3$ has a partial transpose whose four eigenvalues are $(1+p)/4$, $(1+p)/4$, $(1+p)/4$, and $(1-3p)/4$. The negativity is the sum of the absolute values of the negative eigenvalues. What is the negativity $N$?
  advanced · physics · Negativity
• True / Falsecorenew
  A two-qubit Werner state with parameter $p = 1/2$ has a partial transpose $\rho^{T_B}$ whose smallest eigenvalue is $(1 - 3p)/4$. By the Peres-Horodecki (PPT) criterion, a two-qubit state is entangled if and only if $\rho^{T_B}$ has a negative eigenvalue. Is this state entangled?
  advanced · physics · The PPT Criterion
• True / Falsecorenew
  In a 2x2 (two-qubit) or 2x3 system, bound entanglement exists: there are states that are entangled yet have zero distillable entanglement.
  advanced · physics · Bound Entanglement
• True / Falsecorenew
  The GHZ state $(|000\rangle + |111\rangle)/\sqrt{2}$ and the W state $(|001\rangle + |010\rangle + |100\rangle)/\sqrt{3}$ belong to the same SLOCC equivalence class, so one can be converted into the other by invertible local operations with nonzero probability.
  advanced · physics · Multipartite Entanglement
• Numericcorenew
  After running this circuit, what is the probability of measuring |000⟩?
  advanced · physics · GHZ vs W States
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 1476), how many shots land on |000⟩? Estimate P(|000⟩) × 1024 and pick the closest option.
  advanced · physics · GHZ vs W States
• Numericcorenew
  For this circuit, what is the probability that measuring all 3 qubits yields exactly |111⟩ (qubit 0 leftmost)?
  advanced · physics · GHZ vs W States
• Multiple choicecorenew
  The CKW monogamy inequality for three qubits, using the tangle (squared concurrence), states $C_{AB}^2 + C_{AC}^2 \le C_{A(BC)}^2$. For the GHZ and W states, what are the values of the three-tangle $\tau_{ABC} = C_{A(BC)}^2 - C_{AB}^2 - C_{AC}^2$?
  advanced · physics · Monogamy of Entanglement
• Multiple choicecorenew
  By Nielsen's theorem, a pure bipartite state $|\psi\rangle$ can be deterministically converted to $|\phi\rangle$ by LOCC if and only if their sorted Schmidt-coefficient vectors satisfy which relation?
  advanced · physics · LOCC and Entanglement
• Multiple choicecorenew
  For the pure state $|\psi\rangle = \cos(\theta)|00\rangle + \sin(\theta)|11\rangle$ with $\theta = \pi/8$, the concurrence is $C = |\sin(2\theta)|$. The entanglement of formation is $E_F = h(x)$, where $x = (1 + \sqrt{1 - C^2})/2$ and $h(x) = -x\log_2(x) - (1-x)\log_2(1-x)$ is the binary entropy. What is $E_F$, in ebits?
  advanced · physics · Checkpoint: Measure Entanglement
• Multiple choicecorenew
  Recall the CHSH quantity S = E(0,0) + E(0,1) + E(1,0) - E(1,1) with binary +-1 outcomes. What value does S take for any local hidden-variable theory versus the maximum reachable by quantum mechanics on a maximally entangled (singlet) pair?
  advanced · physics · Recap: Bell Nonlocality
• Multiple choicecorenew
  In the CHSH nonlocal game, referees send Alice bit x and Bob bit y (each uniform in {0,1}); the players win if a XOR b = x AND y, where a, b are their answers. Using only deterministic classical strategies (no shared entanglement, no randomness needed since randomness cannot beat the best deterministic plan), what is the maximum probability of winning?
  advanced · physics · Nonlocal Games
• Multiple choicecorenew
  In the CHSH game the players share a Bell pair and measure along angles set by their inputs: Alice uses $\alpha = \{0, \pi/2\}$ for $x = \{0,1\}$, Bob uses $\beta = \{\pi/4, -\pi/4\}$ for $y = \{0,1\}$. For inputs $(x,y)$ the probability their outcomes agree is $\cos^2((\alpha_x - \beta_y)/2)$. They need agreement unless $x$ AND $y = 1$ (then they need disagreement). Averaging the success probability over the 4 equally likely inputs, what is the overall win probability?
  advanced · physics · The CHSH Game Strategy
• Multiple choicecorenew
  Tsirelson's bound on the CHSH operator follows from $|\langle B\rangle| \le \sqrt{4 + \langle[A_0,A_1]\rangle\langle[B_0,B_1]\rangle}$, where $A_0,A_1$ and $B_0,B_1$ are dichotomic ($\pm 1$) observables. For ideal anticommuting observables each commutator has operator norm 2, so the product term is $2 \cdot 2 = 4$. What is the resulting maximum $|\langle B\rangle|$?
  advanced · physics · Tsirelson’s Bound, Derived
• Multiple choicecorenew
  In CHSH self-testing, observing the maximal Bell value $S = 2\sqrt{2}$ on two untrusted black boxes certifies the underlying state and measurements. To what precision can the shared state be identified from the statistics alone?
  advanced · physics · Device-Independent Certification
• Multiple choicecorenew
  In device-independent certified randomness from the CHSH scenario, the adversary's guessing probability obeys $P_{guess} \le (1/2)(1 + \sqrt{2 - S^2/4})$. At the maximal (Tsirelson) violation $S = 2\sqrt{2}$, how much fresh certified randomness (min-entropy $H_{min} = -\log_2(P_{guess})$) is forced in Alice's single binary outcome?
  advanced · physics · Randomness from Nonlocality
• Numericcorenew
  After running this circuit, what is the probability of measuring |000⟩?
  advanced · physics · The GHZ Paradox
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 7067), how many shots land on |000⟩? Estimate P(|000⟩) × 1024 and pick the closest option.
  advanced · physics · The GHZ Paradox
• Numericcorenew
  For this circuit, what is the probability that measuring all 3 qubits yields exactly |111⟩ (qubit 0 leftmost)?
  advanced · physics · The GHZ Paradox
• True / Falsecorenew
  The Peres-Mermin magic square proof of contextuality is state-independent: the contradiction (+1 = -1) holds for every quantum state and requires no entanglement or spatial separation between measurements.
  advanced · physics · Contextuality
• Multiple choicecorenew
  To close the detection (fair-sampling) loophole in a CHSH Bell test using a maximally entangled state, the detector efficiency eta must exceed which threshold?
  advanced · physics · Loophole-Free Bell Tests
• Multiple choicecorenew
  Two players share the Bell state $|\Phi^+\rangle = (|00\rangle + |11\rangle)/\sqrt{2}$. Alice measures $Z$ (input 0) or $X$ (input 1); Bob measures $(Z+X)/\sqrt{2}$ (input 0) or $(Z-X)/\sqrt{2}$ (input 1). The CHSH value is $S = \langle\Phi^+| (Z\otimes B_0 + Z\otimes B_1 + X\otimes B_0 - X\otimes B_1) |\Phi^+\rangle$, and the optimal win probability of the CHSH game is $1/2 + S/8$. What is this win probability?
  advanced · physics · Checkpoint: Win the CHSH Game
• True / Falsecorenew
  Claim: The one-time pad (ciphertext c = m XOR k with a uniformly random key k used only once and as long as the message) provides information-theoretic perfect secrecy, meaning no amount of computation, classical or quantum, can break a properly used one-time pad.
  advanced · physics · Cryptography Basics
• Numericcorenew
  After running this circuit, what is the probability of measuring |0⟩?
  advanced · physics · The BB84 Protocol
• Numericcorenew
  Run this circuit. What is qubit 0's Bloch X-component — the expectation value ⟨X⟩?
  advanced · physics · The BB84 Protocol
• Predictcorenew
  On the seeded simulator (seed 8758), this circuit runs for 1024 shots. About how many come out |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  advanced · physics · The BB84 Protocol
• Multiple choicecorenew
  In a BB84 intercept-resend attack, Eve measures each qubit in a randomly chosen basis. With probability 1/2 she picks the correct (matching) basis and introduces no error; with probability 1/2 she picks the wrong basis and, after resending, causes Bob to get a wrong bit with probability 1/2. What is the resulting qubit error rate Eve induces (= P(correct)*0 + P(wrong)*1/2)?
  advanced · physics · Eavesdropping and Detection
• True / Falsecorenew
  Claim: The no-cloning theorem follows from the linearity/unitarity of quantum mechanics and implies that a single unitary can clone two states $|\psi\rangle$ and $|\phi\rangle$ only if their overlap $\langle\psi|\phi\rangle$ is 0 or 1; consequently it forbids cloning a non-orthogonal pair such as $|0\rangle$ and $|+\rangle$, whose overlap is $1/\sqrt{2}$.
  advanced · physics · The No-Cloning Basis
• Multiple choicecorenew
  In the E91 protocol the correlation for measurement angles x and y on a singlet is E(x,y) = -cos(x - y). Using Alice's angles a = 0, a' = pi/2 and Bob's angles b = pi/4, b' = 3pi/4, the CHSH quantity is S = E(a,b) - E(a,b') + E(a',b) + E(a',b'). What is |S|?
  advanced · physics · The E91 Protocol
• Multiple choicecorenew
  Security from a Bell violation relies on the monogamy of entanglement, expressed (for CHSH) as $S_{AB}^2 + S_{AE}^2 \le 8$. If Alice and Bob observe the maximal violation $S_{AB} = 2\sqrt{2}$, what does this bound force for Eve's CHSH correlation $S_{AE}$ with Alice?
  advanced · physics · Security from Bell Violation
• Multiple choicecorenew
  In QKD information reconciliation, Alice's key X and Bob's key Y are modeled as a binary symmetric channel with crossover probability Q (the QBER). By Shannon's source-coding bound, what is the minimum public side information Bob needs per key bit to reconstruct X with vanishing error?
  advanced · physics · Information Reconciliation
• Multiple choicecorenew
  Privacy amplification compresses a partly-compromised shared string into a shorter, near-secret key. Which entropy measure does it use to quantify Eve's knowledge, and why is that measure (rather than Shannon entropy) the correct one?
  advanced · physics · Privacy Amplification
• Multiple choicecorenew
  In device-independent QKD (DI-QKD), what is the basis for the security guarantee, given that the internal workings of the devices are untrusted ('black boxes')?
  advanced · physics · Device-Independent QKD
• Multiple choicecorenew
  In Wiesner's quantum money, each banknote qubit is prepared in one of the four BB84 states in a basis unknown to a forger, who can only guess a basis and measure. The lesson works out the probability that a single forged qubit passes the bank's verification as 1/2 + 1/4 = 3/4. What is the probability that an entire forged n-qubit note passes?
  advanced · physics · Quantum Money (Overview)
• Numericcorenew
  After running this circuit, what is the probability of measuring |10⟩?
  advanced · physics · Checkpoint: Run BB84
• Multiple choicecorenew
  Let P be the projector onto a quantum code's codespace and {E_a} be a set of error operators. According to the Knill-Laflamme conditions, a recovery operation that perfectly corrects every error in {E_a} exists if and only if which relation holds for all a, b?
  advanced · physics · The QEC Conditions
• True / Falsecorenew
  Claim: Although physical noise is continuous, if a quantum code corrects a discrete basis of errors (e.g. the Pauli operators) then it automatically corrects every continuous error built from that basis, because the syndrome measurement collapses the coherent superposition of errors onto a single, already-correctable Pauli.
  advanced · physics · Discretization of Errors
• Multiple choicecorenew
  The 5-qubit perfect code is the $[[5,1,d]]$ stabilizer code with cyclic generators $XZZXI$, $IXZZX$, $XIXZZ$, $ZXIXZ$. The distance $d$ is the smallest Pauli weight of an operator that commutes with all four generators (is in $N(S)$) but is not in the stabilizer group $S$. What is $d$?
  advanced · physics · Code Distance
• Multiple choicecorenew
  The quantum Hamming bound for a non-degenerate $[[n,k,d]]$ code that corrects $t$ errors requires $2^k \sum_{j=0}^{t} C(n,j) 3^j \le 2^n$, where the $3^j$ counts weight-$j$ Pauli errors ($X,Y,Z$ per location). For $n = 5, k = 1, t = 1$, what is the left-hand side $2^k \sum_{j=0}^{1} C(5,j) 3^j$?
  advanced · physics · Quantum Hamming Bound
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |000⟩?
  advanced · physics · The Repetition Code Revisited
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 1076). How many of the shots come out |000⟩? Estimate P(|000⟩) × 1024 and pick the closest option.
  advanced · physics · The Repetition Code Revisited
• Numericcorenew
  After this circuit, what is P(|111⟩) for the whole 3-qubit register measured together (qubit 0 leftmost)?
  advanced · physics · The Repetition Code Revisited
• Multiple choicecorenew
  Shor's nine-qubit $[[9,1,3]]$ code concatenates the phase-flip and bit-flip repetition codes and has $n - k = 8$ independent stabilizer generators. How are these 8 generators split between $Z$-type and $X$-type checks?
  advanced · physics · The Nine-Qubit Shor Code (Theory)
• Multiple choicecorenew
  A stabilizer code is called DEGENERATE under which precise condition on two distinct correctable errors E_a and E_b?
  advanced · physics · Degenerate Codes
• Multiple choicecorenew
  For a stabilizer code, the logical Pauli operators Xbar and Zbar are elements of the normalizer (not in S) that must satisfy two commutation requirements. Which pair of requirements correctly captures them?
  advanced · physics · Logical Operators
• Multiple choicecorenew
  The single-qubit depolarizing channel is $D_p(\rho) = (1-p)\rho + (p/3)(X\rho X + Y\rho Y + Z\rho Z)$. With what probability does each of the three Pauli errors $X, Y, Z$ occur, and what makes this channel 'maximally symmetric'?
  advanced · physics · Error Models
• Multiple choicecorenew
  The quantum Singleton bound for an $[[n,k,d]]$ code states $k \le n - 2(d-1)$, equivalently $n - k \ge 2(d-1)$. Why does each unit of distance beyond 1 cost TWO qubits of redundancy, rather than one as in the classical Singleton bound?
  advanced · physics · Bounds on Code Parameters
• Multiple choicecorenew
  Approximate quantum error correction (AQEC) relaxes the exact Knill-Laflamme conditions. The canonical motivating example is amplitude damping (energy decay $|1\rangle \to |0\rangle$) with $\gamma = 1 - \exp(-t/T_1)$. To what order in $\gamma$ do small four-qubit codes correct a single amplitude-damping event?
  advanced · physics · Approximate QEC (Overview)
• Multiple choicecorenew
  Consider the 4-qubit stabilizer code whose stabilizer group is generated by S1 = XXXX and S2 = ZZZZ. The code distance is the minimum Pauli weight of an operator that commutes with all generators (lies in the normalizer N(S)) but is not itself in the stabilizer group S. What is the distance of this code?
  advanced · physics · Checkpoint: Analyze a Code
• Multiple choicecorenew
  The n-qubit Pauli group P_n consists of all tensor products of single-qubit Paulis {I, X, Y, Z} together with an overall phase from {+1, -1, +i, -i}. How many elements does P_n have?
  advanced · physics · The Pauli Group
• True / Falsecorenew
  A candidate stabilizer group on 3 qubits is generated by g1 = Z0 Z1 and g2 = Z1 Z2. A set of Pauli operators forms a valid stabilizer group only if every pair of generators commutes (the group is abelian). Computing the symplectic inner product of g1 and g2 over GF(2), is this generating set abelian?
  advanced · physics · Stabilizer Groups
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |000⟩?
  advanced · physics · Stabilizer States
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 9405). How many of the shots come out |000⟩? Estimate P(|000⟩) × 1024 and pick the closest option.
  advanced · physics · Stabilizer States
• Numericcorenew
  After this circuit, what is P(|111⟩) for the whole 3-qubit register measured together (qubit 0 leftmost)?
  advanced · physics · Stabilizer States
• Multiple choicecorenew
  The 5-qubit perfect code has 4 stabilizer generators, written as rows of a check matrix [X-part | Z-part] over GF(2) with n = 5. The 4 generators are XZZXI, IXZZX, XIXZZ, ZXIXZ. The number of encoded logical qubits is k = n - rank(check matrix). If Gaussian elimination over GF(2) gives the 4 rows full rank 4, what is k?
  advanced · physics · The Check Matrix
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |01011⟩?
  advanced · physics · Measuring Stabilizers
• Multiple choicecorenew
  The Gottesman-Knill theorem concerns circuits using only computational-basis state preparation, Clifford gates (H, S, CNOT), and computational-basis measurement. What does the theorem establish about such circuits?
  advanced · physics · The Gottesman–Knill Theorem
• True / Falsecorenew
  The classical [7,4,3] Hamming code has parity-check matrix H with rows 0001111, 0110011, 1010101. A CSS code can be built from a single classical code when that code is dual-containing, i.e. H H^T = 0 (mod 2). Computing H H^T mod 2 for this H, is every entry zero (so the construction is valid)?
  advanced · physics · CSS Codes
• Multiple choicecorenew
  The Steane code is built from the classical [7,4] Hamming code by taking H_X = H_Z = H, giving three X-type and three Z-type stabilizer generators on 7 physical qubits. How many logical qubits k does it encode?
  advanced · physics · The Steane Code as CSS
• True / Falsecorenew
  On 3 qubits, a stabilizer code has generators g1 = Z0 Z1 and g2 = Z1 Z2. Proposed logical operators are Xbar = X0 X1 X2 and Zbar = Z0. Two Paulis commute iff their symplectic inner product is even. A valid logical pair must commute with every stabilizer generator while anticommuting with each other. Does this pair (Xbar, Zbar) qualify as valid logical operators?
  advanced · physics · Logical Operators in Stabilizer Codes
• True / Falsecorenew
  The encoding circuit for any stabilizer code can always be implemented as a Clifford circuit (built from H, S, and CNOT gates).
  advanced · physics · Encoding Stabilizer Codes
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |000⟩?
  advanced · physics · Graph States
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 3750). How many of the shots come out |000⟩? Estimate P(|000⟩) × 1024 and pick the closest option.
  advanced · physics · Graph States
• Numericcorenew
  After this circuit, what is P(|001⟩) for the whole 3-qubit register measured together (qubit 0 leftmost)?
  advanced · physics · Graph States
• Multiple choicecorenew
  The Laflamme 5-qubit perfect code is defined by the four cyclic generators XZZXI, IXZZX, XIXZZ, ZXIXZ, with logical operators Xbar = XXXXX and Zbar = ZZZZZ. The distance is the minimum Pauli weight of an operator in the normalizer N(S) but not in the stabilizer group S. What is the code distance?
  advanced · physics · Checkpoint: Stabilizer Workout
• Multiple choicecorenew
  The distance d of a quantum error-correcting code is the minimum weight of any logical operator. How many single-qubit errors can a distance-d code correct?
  advanced · physics · Topological Codes Overview
• Multiple choicecorenew
  The toric code is placed on an L x L square lattice with periodic boundary conditions, with one qubit on every edge. How many physical qubits n does it use?
  advanced · physics · The Toric Code
• Multiple choicecorenew
  On the toric code's square lattice, an adjacent star operator A_v (a product of X) and plaquette operator B_p (a product of Z) share exactly 2 edges. X and Z anticommute on each shared edge, so the overall commutation sign is (-1)^(overlap). For overlap = 2, what is this sign?
  advanced · physics · Star and Plaquette Operators
• Multiple choicecorenew
  In the toric code, a Z-error creates a pair of e (charge) anyons on vertices and an X-error creates a pair of m (flux) anyons on plaquettes. What happens to the state when an e anyon is dragged all the way around an m anyon along a closed loop?
  advanced · physics · Anyonic Excitations
• True / Falsecorenew
  In the toric code, a closed Z-string around a contractible loop (the boundary of a region of faces) is a product of the plaquette operators inside that region, and therefore acts as an element of the stabilizer group on the code space.
  advanced · physics · Logical Operators as Loops
• Multiple choicecorenew
  The toric code encodes k = 2 logical qubits. When it is opened into a planar surface-code patch with boundaries (two opposite rough and two opposite smooth boundaries), how many logical qubits does the patch encode?
  advanced · physics · The Planar Surface Code
• Multiple choicecorenew
  A rotated surface code uses $n = d^2$ physical qubits, so the distance is $d = \sqrt{n}$. With $n = 25$ physical qubits, the number of arbitrary errors it can correct is $t = \lfloor (d - 1)/2 \rfloor$. What is $t$?
  advanced · physics · Code Distance on the Lattice
• Multiple choicecorenew
  In minimum-weight perfect matching (MWPM) decoding under independent noise where each qubit errs with small probability p < 1/2, why does choosing the lowest-weight error consistent with the syndrome maximize the success probability?
  advanced · physics · Decoding and Matching
• Multiple choicecorenew
  Below threshold, a surface code's logical error rate scales as P_L = A * (p/pth)^((d+1)/2). Take prefactor A = 0.1, physical error rate p = 0.001, threshold pth = 0.01, and distance d = 5. What is P_L?
  advanced · physics · Error Thresholds
• True / Falsecorenew
  In a 2D color code, the entire Clifford group -- including the Hadamard H, phase S, and CNOT between two code blocks -- can be implemented transversally (as products of single-physical-qubit gates).
  advanced · physics · Color Codes (Overview)
• Multiple choicecorenew
  For the toric code on an $L \times L$ square lattice with $L = 4$: one qubit lives on each edge so $n = 2L^2$, the genus-1 torus encodes $k = 2$ logical qubits, and the code distance is the shortest non-contractible loop $d = L$. What is $n + k + d$?
  advanced · physics · Checkpoint: Toric-Code Logic
• Multiple choicecorenew
  In the fault-tolerance problem, the comforting story of quantum error correction (encode, measure syndrome, recover) relies on one fatal implicit assumption. Which assumption is it?
  advanced · physics · The Fault-Tolerance Problem
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |11⟩?
  advanced · physics · Error Propagation
• Multiple choicecorenew
  For a fault-tolerant gadget built on a distance-3 code, what is the defining error-containment property ('one fault in, at most one error out')?
  advanced · physics · Fault-Tolerant Gadgets
• True / Falsecorenew
  The threshold theorem states that the accuracy threshold p_th is a positive constant that is independent of the size of the computation: below it, an arbitrarily long quantum computation can be made arbitrarily accurate with only polylogarithmic overhead in the target accuracy.
  advanced · physics · The Threshold Theorem
• Multiple choicecorenew
  A code concatenated to L = 3 levels gives logical error p_L = pth * (p/pth)^(2^L), with threshold pth = 1e-2 and physical error p = 1e-3. What is p_L?
  advanced · physics · Concatenated Codes
• Multiple choicecorenew
  A simple recursion gives a fault-tolerance threshold as the root of $2p^2 - 3p + 1 = 0$ that lies strictly between 0 and 1. Using the quadratic formula, what is that threshold $p$?
  advanced · physics · Computing the Threshold
• Multiple choicecorenew
  The Eastin-Knill theorem (2009) places a fundamental restriction on transversal gates. What does it state for any quantum error-detecting code that detects at least one error?
  advanced · physics · Transversal Gates and Limits
• True / Falsecorenew
  In surface-code lattice surgery, merging two patches along a shared smooth boundary performs a non-destructive measurement of the joint logical operator Z-bar_1 Z-bar_2.
  advanced · physics · Lattice Surgery
• Multiple choicecorenew
  A distance-$d$ rotated surface code patch uses $n = 2d^2 - 1$ physical qubits (data plus ancilla). For $d = 11$, how many physical qubits does one logical qubit require?
  advanced · physics · Overhead and Scalability
• True / Falsecorenew
  Observing that a surface-code logical qubit's error rate DECREASES as the code distance grows (e.g. from d=3 to d=5 to d=7) is direct experimental evidence of operating below the fault-tolerance threshold.
  advanced · physics · Roadmaps to Fault Tolerance
• Multiple choicecorenew
  The Bravyi-Kitaev 15-to-1 magic-state distillation protocol (built from the $[[15,1,3]]$ Reed-Muller code) takes 15 input copies of error rate $p$ and outputs one cleaner copy. Approximately what is the output error rate $p_{out}$?
  advanced · physics · Resource States
• Multiple choicecorenew
  A concatenated code suppresses errors doubly-exponentially: after L levels the logical rate is p_L = pth * (p/pth)^(2^L), with threshold pth = 1e-2 and physical rate p = 1e-3. What is the smallest number of concatenation levels L needed to reach p_L <= 1e-15?
  advanced · physics · Checkpoint: Concatenation Math
• Multiple choicecorenew
  Why are magic states needed for fault-tolerant universal quantum computation, given that many codes implement the Clifford group transversally?
  advanced · physics · Why Magic States
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |1⟩?
  advanced · physics · The T Gate and Magic
• Numericcorenew
  After running this circuit, what is the expectation value ⟨X⟩ of qubit 0 (its Bloch X-component)?
  advanced · physics · The T Gate and Magic
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 736). Predict how many give |1⟩. Estimate P(|1⟩) × 1024 and pick the closest option.
  advanced · physics · The T Gate and Magic
• Multiple choicecorenew
  For a single qubit, the six stabilizer states (the eigenstates of X, Y, and Z) form the vertices of which geometric shape inscribed in the Bloch sphere, with magic states lying outside it?
  advanced · physics · Stabilizer States vs Magic
• Multiple choicecorenew
  Magic-state injection teleports a T gate onto $|\psi\rangle = a|0\rangle + b|1\rangle$ with the target $T|\psi\rangle = a|0\rangle + b e^{i\pi/4}|1\rangle$. Both measurement branches are corrected: outcome 0 needs no fix, and outcome 1 is repaired by applying the $S = \mathrm{diag}(1, i)$ gate. After correction, what is the gate fidelity $F = |\langle T\psi | \text{corrected}\rangle|^2$ of the worse of the two branches?
  advanced · physics · Magic-State Injection
• Multiple choicecorenew
  A noisy magic state is the depolarizing mixture $\rho = (1 - p)|T\rangle\langle T| + p(I/2)$. Its infidelity to the ideal $|T\rangle$ is $1 - \langle T|\rho|T\rangle$, where $\langle T|(I/2)|T\rangle = 1/2$. For $p = 0.08$, what is the input error $1 - \langle T|\rho|T\rangle$?
  advanced · physics · Noisy Magic States
• Multiple choicecorenew
  In the Bravyi-Kitaev 15-to-1 magic-state distillation protocol, to leading order the output error relates to the input error as which of the following?
  advanced · physics · Distillation Protocols
• Multiple choicecorenew
  Magic-state distillation with eps_out = 35 * eps_in^3 only improves the state when eps_out < eps_in. The break-even threshold is the fixed point where eps_out = eps_in. What is that threshold eps*?
  advanced · physics · Distillation Thresholds
• Multiple choicecorenew
  Using 15-to-1 distillation ($\epsilon \to 35\epsilon^3$, costing 15 raw states per round), how many raw magic states are consumed per output to push $\epsilon$ from 1e-3 down to a target of 1e-12? Each round multiplies the raw cost by 15.
  advanced · physics · Resource Cost of Distillation
• Multiple choicecorenew
  In the resource theory of stabilizer computation, what is the defining faithfulness property of a magic monotone $M(\rho)$?
  advanced · physics · Magic Monotones (Overview)
• Multiple choicecorenew
  A single round of the 15-to-1 magic-state distillation protocol maps an input error rate eps_in to eps_out = 35 * eps_in^3. For eps_in = 0.01, what is the output error rate eps_out?
  advanced · physics · Checkpoint: Distillation Math
• True / Falsecorenew
  In the one-way (MBQC) model, after the cluster state is prepared each qubit is measured exactly once and then discarded, so the entanglement is consumed and the process cannot be undone — which is the origin of the name 'one-way'.
  advanced · physics · The One-Way Model
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |00⟩?
  advanced · physics · Cluster States
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 5928). How many of the shots come out |00⟩? Estimate P(|00⟩) × 1024 and pick the closest option.
  advanced · physics · Cluster States
• Numericcorenew
  After this circuit, what is P(|01⟩) for the whole 2-qubit register measured together (qubit 0 leftmost)?
  advanced · physics · Cluster States
• Multiple choicecorenew
  In the one-bit teleportation gadget, an input $|\psi\rangle$ on wire 1 is entangled with an ancilla $|+\rangle$ on wire 2 via controlled-$Z$, then wire 1 is measured in the $X$ basis with outcome $s$. What state does wire 2 hold?
  advanced · physics · Computing by Measurement
• Multiple choicecorenew
  In the measurement calculus, a measurement pattern is built from exactly four primitive commands. Which set correctly lists them?
  advanced · physics · Measurement Patterns
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|0⟩)?
  advanced · physics · Feed-Forward Corrections
• Numericcorenew
  This circuit prepares qubit 0 in some state. What expectation value ⟨Y⟩ does it have along the Bloch Y-axis?
  advanced · physics · Feed-Forward Corrections
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 8961). How many of the shots come out |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  advanced · physics · Feed-Forward Corrections
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |00⟩?
  advanced · physics · Simulating MBQC Gates
• Numericcorenew
  After running this circuit, what is the expectation value ⟨X⟩ of qubit 0 (its Bloch X-component)?
  advanced · physics · Simulating MBQC Gates
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 2432). Predict how many give |00⟩. Estimate P(|00⟩) × 1024 and pick the closest option.
  advanced · physics · Simulating MBQC Gates
• Numericcorenew
  For this circuit, what is the probability that measuring all 2 qubits yields exactly |01⟩ (qubit 0 leftmost)?
  advanced · physics · Simulating MBQC Gates
• Multiple choicecorenew
  The lesson states that the one-way (MBQC) model and the circuit model are equivalent in computational power. Precisely which complexity class does MBQC solve?
  advanced · physics · Equivalence to the Circuit Model
• Multiple choicecorenew
  For a graph state |G> on graph G = (V, E), the stabilizer generators are read directly off the graph. What is the generator K_a associated with vertex a (where N(a) is the neighborhood of a)?
  advanced · physics · Graph States and MBQC
• Multiple choicecorenew
  In the Raussendorf-Harrington-Goyal topological (3D cluster) approach to fault-tolerant MBQC, what role does the third dimension of the three-dimensional cluster state play?
  advanced · physics · Fault Tolerance in MBQC
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|0000⟩)?
  advanced · physics · Checkpoint: Measure to Compute
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 116). Predict how many give |0000⟩. Estimate P(|0000⟩) × 1024 and pick the closest option.
  advanced · physics · Checkpoint: Measure to Compute
• Numericcorenew
  After running this circuit, what is the probability that all 4 qubits together read |0001⟩ (qubit 0 leftmost)?
  advanced · physics · Checkpoint: Measure to Compute
• Multiple choicecorenew
  The lesson defines topological order partly through the entanglement entropy of a region A with smooth boundary: $S(A) = \alpha|\partial A| - \gamma + \dots$ . What distinguishes a topologically ordered phase from a trivial phase in terms of the universal constant $\gamma$?
  advanced · physics · What Is Topological Order
• Multiple choicecorenew
  Why are anyons possible in two spatial dimensions but not in three or more? The lesson attributes the difference to the fundamental group of the configuration space of n distinct points.
  advanced · physics · Anyons: Beyond Bosons and Fermions
• Multiple choicecorenew
  In the toric code, the electric charge e (a Z-string endpoint) and magnetic flux m (an X-string endpoint) are each individually bosons. What phase does the state acquire when an m is carried on a closed loop fully around a single e?
  advanced · physics · Abelian Anyons
• Multiple choicecorenew
  For Ising anyons with sectors $\{1, \sigma, \psi\}$, the lesson gives the quantum dimensions $d_\sigma = \sqrt{2}$, $d_\psi = 1$, $d_1 = 1$. What is the total quantum dimension $D = \sqrt{\sum_a d_a^2}$?
  advanced · physics · Non-Abelian Anyons
• Multiple choicecorenew
  In topological quantum computation, why are gates implemented by braiding non-abelian anyons intrinsically protected from small errors?
  advanced · physics · Braiding and Computation
• Multiple choicecorenew
  Fibonacci anyons have two charges, vacuum and tau, with fusion rules vacuum x tau = tau and tau x tau = vacuum + tau. Starting from a single tau and fusing in 5 more taus one at a time (6 taus total), how many distinct fusion trees end in the vacuum channel?
  advanced · physics · Fusion Rules
• Multiple choicecorenew
  In Kitaev's exactly solvable chain (fine-tuned topological limit: chemical potential mu=0, hopping = pairing = t), the two unpaired Majorana modes left at the ends of the wire are combined into a single nonlocal fermionic mode f. Why can no local perturbation read or flip the qubit stored in its occupation n_f?
  advanced · physics · Majorana Zero Modes
• Multiple choicecorenew
  According to the lesson on topological quantum computation, which statement correctly contrasts Ising/Majorana braiding with Fibonacci braiding?
  advanced · physics · Topological Quantum Computation
• Multiple choicecorenew
  In a topologically ordered system with a uniform energy gap Delta, the lesson states that the probability of real (thermal) excitation of a stray anyon is suppressed by which factor (k_B is Boltzmann's constant, T the temperature)?
  advanced · physics · Protection from Errors
• True / Falsecorenew
  True or False: According to the lesson, a quantized zero-bias conductance peak of $2e^2/h$ in a semiconductor nanowire is, by itself, conclusive proof that genuine Majorana zero modes are present.
  advanced · physics · Realizations and Challenges
• Multiple choicecorenew
  Ising ($\sigma$) anyons have fusion rules: $\text{vacuum} \times \sigma = \sigma$, $\sigma \times \sigma = \text{vacuum} + \psi$, and $\psi \times \sigma = \sigma$. Fuse 6 $\sigma$ anyons one at a time from left to right (start: a single $\sigma$, then bring in 5 more). How many distinct fusion trees end with total charge = vacuum?
  advanced · physics · Checkpoint: Fusion and Braiding
• Multiple choicecorenew
  According to the lesson, what does the 'NP' in the complexity class NP stand for?
  advanced · physics · Computational Complexity Basics
• Multiple choicecorenew
  BQP is defined with acceptance probability thresholds 2/3 (for x in L) and 1/3 (for x not in L). The lesson explains why this particular gap is not special. What technique shows that any inverse-polynomial gap defines the same class?
  advanced · physics · The Class BQP
• Multiple choicecorenew
  The lesson establishes the proven containment chain P subset of BPP subset of BQP subset of PSPACE subset of EXP. The genuinely nontrivial upper bound is BQP subset of PSPACE. What is the key idea behind that proof?
  advanced · physics · BQP vs Classical Classes
• Multiple choicecorenew
  In the class QMA (Quantum Merlin-Arthur), how does the witness (proof) differ from the witness in the classical class NP?
  advanced · physics · The Class QMA
• True / Falsecorenew
  True or False: The lesson states that the local Hamiltonian problem is QMA-complete, making it the QMA analogue of what SAT is for NP.
  advanced · physics · The Local Hamiltonian Problem
• Multiple choicecorenew
  Grover search over $N = 16$ items with $M = 1$ marked item uses rotation angle $\theta = \arcsin(\sqrt{1/16})$. After the optimal $r = \mathrm{round}(\pi/(4\theta) - 1/2) = 3$ iterations, the success probability is $\sin^2((2r+1)\theta)$. What is this probability?
  advanced · physics · Quantum Query Complexity
• Multiple choicecorenew
  The lesson states that the modern negative-weight (general) adversary bound ADV-plus-minus(g) has a special status among quantum query lower-bound techniques. What is that status?
  advanced · physics · Lower Bounds and Adversary Methods
• Multiple choicecorenew
  The lesson describes the linear cross-entropy benchmark (linear XEB) used to verify random-circuit-sampling supremacy experiments. The XEB fidelity is normalized so that a perfect ideal sampler scores 1. What does a device producing only uniform noise score?
  advanced · physics · Quantum Supremacy
• Multiple choicecorenew
  In boson sampling, the collision-free output amplitudes are permanents of n-by-n submatrices of the interferometer unitary. Why does this make sampling classically hard, whereas the analogous determinant-based problem would be easy?
  advanced · physics · Boson Sampling
• Multiple choicecorenew
  The lesson distinguishes two simulation tasks. Simulating real-time quantum dynamics (approximating e^(-iHt)|psi_0> for a local Hamiltonian H) has which complexity, according to the lesson?
  advanced · physics · Complexity of Simulation
• Multiple choicecorenew
  An unstructured search over $N = 2^6 = 64$ items has a single marked item. Using Grover's algorithm, the optimal number of oracle queries is $\mathrm{round}((\pi/4)\sqrt{N} - 1/2)$. How many quantum oracle queries does this give for $N = 64$?
  advanced · physics · Checkpoint: Query Complexity
• Multiple choicecorenew
  For a system S coupled to a bath B, the total Hamiltonian is written as H = H_S (x) 1_B + 1_S (x) H_B + H_SB. What happens to the system S if the interaction term H_SB equals zero?
  advanced · physics · System–Environment Coupling
• Multiple choicecorenew
  Consider the Bell state $|\Phi^+\rangle = (1/\sqrt{2})(|00\rangle + |11\rangle)$ of system S and environment E. Trace out E to get the reduced density matrix $\rho_S$, then compute the purity $P = \mathrm{Tr}(\rho_S^2)$. What is $P$?
  advanced · physics · The Reduced Density Matrix
• Multiple choicecorenew
  In the canonical decoherence model, a system qubit $\alpha|0\rangle + \beta|1\rangle$ becomes entangled with an environment so the reduced density matrix $\rho_S$ has off-diagonal coherences multiplied by the decoherence factor $D = \langle e_0|e_1\rangle$. What happens to $\rho_S$ when the two branches drive the environment into orthogonal states ($D = 0$)?
  advanced · physics · Decoherence
• Multiple choicecorenew
  In the Born-Markov approximation, the 'Markov' (no-memory) assumption is expressed as a separation of two timescales: the bath correlation time tau_B and the system relaxation time tau_S. Which condition expresses Markovianity?
  advanced · physics · The Born–Markov Approximation
• Multiple choicecorenew
  Under amplitude damping the excited-state population obeys d(rho_11)/dt = -gamma rho_11, so rho_11(t) = rho_11(0) e^(-gamma t). With gamma = 1.5 (per unit time) and rho_11(0) = 1.0, what is rho_11 at t = 2.0?
  advanced · physics · The Lindblad Master Equation
• Multiple choicecorenew
  Under pure dephasing the off-diagonal density-matrix element obeys d(rho_01)/dt = -gamma rho_01, so rho_01(t) = rho_01(0) e^(-gamma t). With gamma = 0.8 (per unit time) and rho_01(0) = 0.5, what is rho_01 at t = 1.25?
  advanced · physics · Jump Operators
• Multiple choicecorenew
  For a qubit with relaxation time T1 = 80 us and pure-dephasing time Tphi = 120 us, the coherence time satisfies 1/T2 = 1/(2 T1) + 1/Tphi. What is T2, in microseconds?
  advanced · physics · Relaxation and Dephasing (T₁, T₂)
• Multiple choicecorenew
  A two-level system has excitation rate gamma_up and relaxation rate gamma_down. Its steady-state excited-state population is p1* = gamma_up/(gamma_up + gamma_down) = r/(1+r) with r = gamma_up/gamma_down. For r = 0.25, what is p1*?
  advanced · physics · Steady States
• Multiple choicecorenew
  Quantum trajectory (Monte Carlo wavefunction) methods are computationally attractive because they propagate a pure state vector instead of a density matrix. For a system of n qubits, how do the sizes of these two objects compare?
  advanced · physics · Quantum Trajectories (Overview)
• Multiple choicecorenew
  Two initial states of an open quantum system are compared using their trace distance D(rho1, rho2), which measures how distinguishable they are. Under any Markovian (CP-divisible) evolution, how can this distinguishability change over time, and what does a temporary INCREASE signal?
  advanced · physics · Non-Markovian Dynamics (Overview)
• Multiple choicecorenew
  A qubit has T1 = 50 us and pure-dephasing time Tphi = 75 us, with 1/T2 = 1/(2 T1) + 1/Tphi. Starting from <X>(0) = 1, the coherence decays as <X>(t) = e^(-t/T2). What is <X> at t = 30 us?
  advanced · physics · Checkpoint: Solve a Master Equation
• True / Falsecorenew
  In quantum scattering theory, the differential cross-section equals the squared modulus of the scattering amplitude: $d\sigma/d\Omega = |f(\theta, \phi)|^2$.
  advanced · physics · Scattering Theory Basics
• Multiple choicecorenew
  For a purely s-wave (isotropic) scatterer with wavenumber $k = 2$ and phase shift $\delta_0 = 0.5$ rad, the scattering amplitude has $|f|^2 = (1/k^2) \sin^2(\delta_0)$, and the total cross section is $\sigma = 4 \pi |f|^2$. What is $\sigma$?
  advanced · physics · The Scattering Amplitude
• Multiple choicecorenew
  In partial-wave analysis of scattering from a central potential, what happens at LOW energy (k*a << 1, where a is the potential range), and why?
  advanced · physics · Partial Wave Analysis
• Multiple choicecorenew
  For low-energy s-wave scattering off a hard sphere of radius $a = 1$, the phase shift is $\delta_0 = -k a$ and the partial cross section is $\sigma_0 = (4\pi / k^2) \sin^2(\delta_0)$. With wavenumber $k = 0.2$, what is $\sigma_0$?
  advanced · physics · Phase Shifts
• Multiple choicecorenew
  In the Born approximation, a Yukawa-type potential gives scattering amplitude $f = -\beta / (\mu^2 + q^2)$, where $q = 2 k \sin(\theta/2)$ is the momentum transfer. With $\beta = 1$, $\mu = 1$, $k = 1$, and backscattering at $\theta = \pi$, what is $f$?
  advanced · physics · The Born Approximation
• Multiple choicecorenew
  The Klein-Gordon equation, despite its interpretational difficulties with negative energies and negative probability densities, correctly describes which kind of particle?
  advanced · physics · The Klein–Gordon Equation
• Multiple choicecorenew
  Dirac sought a relativistic wave equation that is FIRST order in the time derivative. For the resulting Dirac equation, what is the time component of the conserved current $j^0 = \rho$, and what desirable property does it have?
  advanced · physics · The Dirac Equation
• Multiple choicecorenew
  According to relativistic quantum mechanics, what is the relationship between a particle and its antiparticle?
  advanced · physics · Antiparticles and Spin
• Multiple choicecorenew
  In second quantization, a free field is expanded in ladder operators a and a-dagger for each mode. Reinterpreting the oscillator quantum number n as a particle number, what roles do these operators play?
  advanced · physics · Toward Quantum Field Theory
• Multiple choicecorenew
  In the first Born approximation a uniform spherical potential of strength V0 and radius a gives the scattering amplitude f = -V0 [sin(q a) - q a cos(q a)] / q^3. With V0 = 1, a = 1, and momentum transfer q = 1, what is f?
  advanced · physics · Checkpoint: Scattering Calculations
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |0⟩?
  advanced · programming · Gate Decomposition
• Numericcorenew
  For qubit 0 after this circuit, what is ⟨X⟩, the X-axis projection of its Bloch vector?
  advanced · programming · Gate Decomposition
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 2117), how many shots land on |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  advanced · programming · Gate Decomposition
• Multiple choicecorenew
  According to the lesson, why are the Clifford gates (generated by H, S, and CNOT) NOT universal for quantum computation on their own?
  advanced · programming · Universal Gate Sets
• Multiple choicecorenew
  The lesson's standard (textbook) decomposition of a Toffoli (CCX) on three clean qubits into one- and two-qubit gates uses how many T-type gates and CNOTs?
  advanced · programming · Synthesizing Multi-Qubit Gates
• Multiple choicecorenew
  The Solovay-Kitaev theorem bounds how long a gate sequence from a dense, inverse-closed single-qubit set must be to approximate a target U in SU(2) to accuracy eps. How does that sequence length L scale with eps?
  advanced · programming · Solovay–Kitaev (Overview)
• Multiple choicecorenew
  A device provides CNOT but not a native SWAP. Using the circuit identity from the lesson, how is SWAP(0,1) realized with CNOTs only?
  advanced · programming · Circuit Identities and Simplification
• True / Falsecorenew
  True or false: On near-term hardware, inserting a SWAP to route qubits is essentially free, because a SWAP is a single native operation that carries no extra error.
  advanced · programming · Qubit Routing and Connectivity
• True / Falsecorenew
  True or false: In a transpiler pipeline, the optimization stage (cancellation, rotation merging) must run AFTER routing, because routing inserts SWAPs (three CNOTs each) that the optimizer then needs to clean up.
  advanced · programming · Transpilation Passes
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|00⟩)?
  advanced · programming · Optimizing Circuit Depth
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 4739). Predict how many give |00⟩. Estimate P(|00⟩) × 1024 and pick the closest option.
  advanced · programming · Optimizing Circuit Depth
• Numericcorenew
  After running this circuit, what is the probability that all 2 qubits together read |11⟩ (qubit 0 leftmost)?
  advanced · programming · Optimizing Circuit Depth
• Multiple choicecorenew
  In approximate compilation, a circuit's noise error grows with its gate count: if each two-qubit gate has error rate p and the circuit uses g' of them, the noise contributes about g'*p. If you accept a unitary approximation error eps to shrink the gate count, the lesson models the total error as roughly which expression?
  advanced · programming · Approximate Compilation
• Multiple choicecorenew
  On a CZ-native device you must re-express CX(0->1) using only CZ and single-qubit gates. Using the identity $H Z H = X$, which sequence implements CX(0->1) (control 0, target 1)?
  advanced · programming · Checkpoint: Compile to a Gate Set
• Multiple choicecorenew
  The lesson explains that the naive 'store a qubit three times and majority vote' strategy fails for quantum error correction. What is the fundamental reason?
  advanced · programming · Recap: Why Error Correction
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|000000000⟩)?
  advanced · programming · The Shor Nine-Qubit Code
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 3986). Predict how many give |000000000⟩. Estimate P(|000000000⟩) × 1024 and pick the closest option.
  advanced · programming · The Shor Nine-Qubit Code
• Numericcorenew
  After running this circuit, what is the probability that all 9 qubits together read |000000111⟩ (qubit 0 leftmost)?
  advanced · programming · The Shor Nine-Qubit Code
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |000⟩?
  advanced · programming · Encoding Circuits
• Predictcorenew
  On the seeded simulator (seed 7019), this circuit runs for 1024 shots. About how many come out |000⟩? Estimate P(|000⟩) × 1024 and pick the closest option.
  advanced · programming · Encoding Circuits
• Numericcorenew
  Run this circuit, then measure all 3 qubits at once. What is the probability of the joint outcome |111⟩ (qubit 0 leftmost)?
  advanced · programming · Encoding Circuits
• Numericcorenew
  After running this circuit, what is the probability of measuring |10010⟩?
  advanced · programming · Syndrome Extraction Circuits
• Multiple choicecorenew
  A 3-qubit bit-flip code uses two parity-check (stabilizer) measurements $Z_0 Z_1$ and $Z_1 Z_2$, giving syndrome $(s_1, s_2)$. The lookup table is: $(1,0)$ -> error on qubit 0, $(1,1)$ -> error on qubit 1, $(0,1)$ -> error on qubit 2, $(0,0)$ -> no error. Which qubit index does syndrome $(1,1)$ identify as flipped?
  advanced · programming · Decoding from Syndromes
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|010⟩)?
  advanced · programming · Correcting Arbitrary Single-Qubit Errors
• Numericcorenew
  This circuit prepares qubit 0 in some state. What expectation value ⟨X⟩ does it have along the Bloch X-axis?
  advanced · programming · Correcting Arbitrary Single-Qubit Errors
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 1827). How many of the shots come out |010⟩? Estimate P(|010⟩) × 1024 and pick the closest option.
  advanced · programming · Correcting Arbitrary Single-Qubit Errors
• Numericcorenew
  Run this circuit, then measure all 3 qubits at once. What is the probability of the joint outcome |110⟩ (qubit 0 leftmost)?
  advanced · programming · Correcting Arbitrary Single-Qubit Errors
• Multiple choicecorenew
  The lesson presents the five-qubit code, denoted [[5,1,3]]. What does this code achieve that no smaller code can?
  advanced · programming · The Five-Qubit Code
• Multiple choicecorenew
  The seven-qubit Steane code [[7,1,3]] is a CSS code built from the classical [7,4,3] Hamming code. Why is its CSS structure considered powerful for fault tolerance?
  advanced · programming · The Steane Code
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |111⟩?
  advanced · programming · Logical Operations on Encoded Qubits
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |010⟩?
  advanced · programming · Measuring Code Performance
• Numericcorenew
  After running this circuit, what is the probability of measuring |001⟩?
  advanced · programming · Checkpoint: Run a QEC Cycle
• Numericcorenew
  Run this circuit. What is qubit 0's Bloch X-component — the expectation value ⟨X⟩?
  advanced · programming · Checkpoint: Run a QEC Cycle
• Predictcorenew
  On the seeded simulator (seed 1366), this circuit runs for 1024 shots. About how many come out |001⟩? Estimate P(|001⟩) × 1024 and pick the closest option.
  advanced · programming · Checkpoint: Run a QEC Cycle
• Numericcorenew
  After this circuit, what is P(|101⟩) for the whole 3-qubit register measured together (qubit 0 leftmost)?
  advanced · programming · Checkpoint: Run a QEC Cycle
• Multiple choicecorenew
  In the rotated surface-code layout, a patch encoding one logical qubit at distance $d$ uses $d^2$ data qubits and $d^2 - 1$ ancilla qubits. How many physical qubits does the smallest useful patch, $d = 3$, need in total?
  advanced · programming · The Surface-Code Lattice
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |00⟩?
  advanced · programming · Plaquette and Vertex Stabilizers
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 3949). How many of the shots come out |00⟩? Estimate P(|00⟩) × 1024 and pick the closest option.
  advanced · programming · Plaquette and Vertex Stabilizers
• Numericcorenew
  After this circuit, what is P(|11⟩) for the whole 2-qubit register measured together (qubit 0 leftmost)?
  advanced · programming · Plaquette and Vertex Stabilizers
• Multiple choicecorenew
  To measure the Z-type plaquette check $Z_0 Z_1$ with a fresh ancilla a prepared in $|0\rangle$, two CNOTs are applied: CNOT(0->a) then CNOT(1->a). What value does the ancilla hold afterward, for data $|q_0 q_1\rangle$?
  advanced · programming · Stabilizer Measurement Cycles
• Multiple choicecorenew
  The lesson stresses that a surface-code decoder does not act on the raw value of a stabilizer check, but on a 'detection event'. How is the detection event for a check defined across rounds?
  advanced · programming · Detecting Errors as Syndromes
• Multiple choicecorenew
  Minimum-weight perfect matching (MWPM) decodes a surface code by pairing up 'detection events'. For Z-type checks, how many detection events does a single X error on a bulk (interior) data qubit produce?
  advanced · programming · Matching Decoders (Overview)
• Multiple choicecorenew
  On a surface-code patch, the logical operators Xbar and Zbar are strings stretching between opposite boundaries. The lesson explains they anticommute (Xbar Zbar = -Zbar Xbar), exactly like single-qubit X and Z. What geometric fact makes them anticommute?
  advanced · programming · Logical Qubits from Boundaries
• Multiple choicecorenew
  A quantum error-correcting code has code distance d = 3. The number of arbitrary single-qubit errors it can correct is t = floor((d - 1) / 2). What is t?
  advanced · programming · Code Distance and Errors
• Multiple choicecorenew
  According to the threshold theorem for the surface code, what happens to the logical error rate when the per-operation physical error rate p is BELOW the threshold p_th and you increase the code distance d?
  advanced · programming · The Threshold Theorem (Applied)
• Multiple choicecorenew
  A rotated planar surface-code patch of distance $d$ uses $n = 2 d^2 - 1$ physical qubits (data plus syndrome). For $d = 3$, how many physical qubits does one logical qubit require?
  advanced · programming · Resource Costs of the Surface Code
• Multiple choicecorenew
  On a 3-data-qubit chain with checks S1 = Z0 Z1 and S2 = Z1 Z2, a single bit-flip X strikes the MIDDLE data qubit (qubit 1). What syndrome (S1, S2) do the two checks report, and why?
  advanced · programming · Checkpoint: One Surface-Code Round
• Multiple choicecorenew
  What is the defining requirement that makes a logical gate fault-tolerant, as stated in this lesson?
  advanced · programming · Fault-Tolerance Goals
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |000⟩?
  advanced · programming · Transversal Gates
• Predictcorenew
  On the seeded simulator (seed 6115), this circuit runs for 1024 shots. About how many come out |000⟩? Estimate P(|000⟩) × 1024 and pick the closest option.
  advanced · programming · Transversal Gates
• Numericcorenew
  Run this circuit, then measure all 3 qubits at once. What is the probability of the joint outcome |111⟩ (qubit 0 leftmost)?
  advanced · programming · Transversal Gates
• Numericcorenew
  After running this circuit, what is the probability of measuring |1⟩?
  advanced · programming · The Clifford Group
• Multiple choicecorenew
  What does the Gottesman–Knill theorem establish about a quantum circuit that starts in a computational-basis state, uses only Clifford gates (H, S, CNOT), and measures in the computational basis?
  advanced · programming · Why Clifford Is Not Universal
• Multiple choicecorenew
  The T gate is T = diag(1, e^{i pi/4}). Applying T twice on a single qubit produces which Clifford gate?
  advanced · programming · Adding the T Gate
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |1⟩?
  advanced · programming · Magic-State Injection
• Numericcorenew
  For qubit 0 after this circuit, what is ⟨X⟩, the X-axis projection of its Bloch vector?
  advanced · programming · Magic-State Injection
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 3956), how many shots land on |1⟩? Estimate P(|1⟩) × 1024 and pick the closest option.
  advanced · programming · Magic-State Injection
• Numericcorenew
  After running this circuit, what is the probability of measuring |0⟩?
  advanced · programming · Approximating Rotations with Clifford+T
• Numericcorenew
  Run this circuit. What is qubit 0's Bloch Y-component — the expectation value ⟨Y⟩?
  advanced · programming · Approximating Rotations with Clifford+T
• Predictcorenew
  On the seeded simulator (seed 1334), this circuit runs for 1024 shots. About how many come out |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  advanced · programming · Approximating Rotations with Clifford+T
• Multiple choicecorenew
  On a fault-tolerant architecture the T gate is the costly non-Clifford resource. A Toffoli (CCX) gate is decomposed into the Clifford+T basis. How many T gates does the standard exact Toffoli decomposition use (its T-count)?
  advanced · programming · T-Count and T-Depth
• True / Falsecorenew
  In realistic fault-tolerant resource estimates, magic-state distillation dominates both the physical qubit count and the runtime, which is why T-count is the headline number.
  advanced · programming · Logical Circuit Compilation
• Multiple choicecorenew
  Using RZ(pi/2) = S and RZ(pi/4) = T (up to global phase), and the fact that same-axis z-rotations add their angles, which Clifford+T word realizes RZ(3*pi/4)?
  advanced · programming · Checkpoint: Synthesize a Logical Gate
• Multiple choicecorenew
  To reach accuracy $\epsilon$ over time $t$, first-order Trotter needs $r = O(t^2/\epsilon)$ steps. How does the accuracy ($\epsilon$) dependence of LCU, qubitization, and QSP compare?
  advanced · programming · Beyond Trotter
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |0⟩?
  advanced · programming · Higher-Order Product Formulas
• Numericcorenew
  After running this circuit, what is the expectation value ⟨Z⟩ of qubit 0 (its Bloch Z-component)?
  advanced · programming · Higher-Order Product Formulas
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 1199). Predict how many give |0⟩. Estimate P(|0⟩) × 1024 and pick the closest option.
  advanced · programming · Higher-Order Product Formulas
• Multiple choicecorenew
  A first-order Lie-Trotter step for $H = X + Z$ has additive error bounded by $(t^2 / 2) \cdot ||[X, Z]||$, where $[X, Z] = -2iY$ so $||[X, Z]|| = 2$. For a single step of duration $t = 0.1$, what is this error bound?
  advanced · programming · Error Scaling of Product Formulas
• Multiple choicecorenew
  In the LCU technique, what is the order of the three operations applied to the ancilla and system before post-selecting on the ancilla returning to $|0\rangle$?
  advanced · programming · Linear Combination of Unitaries (LCU)
• Multiple choicecorenew
  In qubitization, given an $(\alpha, m)$-block-encoding of $H$, the walk operator $W$ has two eigenvalues $e^{\pm i \theta_k}$ within each invariant 2D block for an eigenvalue $E_k$ of $H$. What is $\theta_k$?
  advanced · programming · Qubitization (Overview)
• Multiple choicecorenew
  In single-qubit Quantum Signal Processing, interleaving d signal rotations W(a) with d+1 tunable Z-rotations e^{i phi_k Z}, what is the top-left entry of the resulting unitary U_phi(a) as a function of the signal a?
  advanced · programming · Quantum Signal Processing (Overview)
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|10⟩)?
  advanced · programming · Simulating the Heisenberg Model
• Numericcorenew
  This circuit prepares qubit 0 in some state. What expectation value ⟨Z⟩ does it have along the Bloch Z-axis?
  advanced · programming · Simulating the Heisenberg Model
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 8550). How many of the shots come out |10⟩? Estimate P(|10⟩) × 1024 and pick the closest option.
  advanced · programming · Simulating the Heisenberg Model
• Numericcorenew
  Run this circuit, then measure all 2 qubits at once. What is the probability of the joint outcome |01⟩ (qubit 0 leftmost)?
  advanced · programming · Simulating the Heisenberg Model
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |0⟩?
  advanced · programming · Measuring Dynamical Quantities
• Numericcorenew
  For qubit 0 after this circuit, what is ⟨X⟩, the X-axis projection of its Bloch vector?
  advanced · programming · Measuring Dynamical Quantities
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 1610), how many shots land on |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  advanced · programming · Measuring Dynamical Quantities
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |0⟩?
  advanced · programming · Time-Dependent Hamiltonians
• Numericcorenew
  After running this circuit, what is the expectation value ⟨Y⟩ of qubit 0 (its Bloch Y-component)?
  advanced · programming · Time-Dependent Hamiltonians
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 9234). Predict how many give |0⟩. Estimate P(|0⟩) × 1024 and pick the closest option.
  advanced · programming · Time-Dependent Hamiltonians
• Multiple choicecorenew
  In the head-to-head comparison of Hamiltonian-simulation methods, how many ancilla qubits do product formulas (Trotter / higher-order Suzuki) require?
  advanced · programming · Comparing Simulation Methods
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |000⟩?
  advanced · programming · Checkpoint: Simulate a Spin Chain
• Predictcorenew
  Of the possible results, which measurement outcome is most probable for this circuit?
  advanced · programming · Checkpoint: Simulate a Spin Chain
• Numericcorenew
  Run this circuit. What is qubit 0's Bloch Z-component — the expectation value ⟨Z⟩?
  advanced · programming · Checkpoint: Simulate a Spin Chain
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 5327). How many of the shots come out |000⟩? Estimate P(|000⟩) × 1024 and pick the closest option.
  advanced · programming · Checkpoint: Simulate a Spin Chain
• Numericcorenew
  After this circuit, what is P(|001⟩) for the whole 3-qubit register measured together (qubit 0 leftmost)?
  advanced · programming · Checkpoint: Simulate a Spin Chain
• Multiple choicecorenew
  This lesson defines the target precision for quantum chemistry as 'chemical accuracy.' What is its conventional value?
  advanced · programming · The Electronic Structure Problem
• Multiple choicecorenew
  The fermionic anticommutation relations include {a_p-dagger, a_q-dagger} = 0. Setting p = q gives a_p-dagger a_p-dagger = 0. Which physical principle does this encode?
  advanced · programming · Second Quantization
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |1010⟩?
  advanced · programming · The Jordan–Wigner Transform
• Multiple choicecorenew
  Compared to the Jordan–Wigner transform, what is the main advantage of the Bravyi–Kitaev (BK) transform for mapping fermionic operators to qubits?
  advanced · programming · The Bravyi–Kitaev Transform
• Multiple choicecorenew
  A 2-qubit molecular Hamiltonian is H = g1*Z0 + g2*Z1 + g3*Z0Z1 + (Y0Y1, X0X1 terms) with g1 = 0.3435, g2 = -0.4347, g3 = 0.5716. Evaluate its energy in the basis state |01> (qubit0 = 0 so <Z0> = +1; qubit1 = 1 so <Z1> = -1; the X0X1 and Y0Y1 terms vanish on a basis state). What is <H>?
  advanced · programming · Molecular Hamiltonians as Paulis
• Numericcorenew
  After running this circuit, what is the probability of measuring |1100⟩?
  advanced · programming · Hartree–Fock Initial States
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |10⟩?
  advanced · programming · VQE for Molecules
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 9209). How many of the shots come out |10⟩? Estimate P(|10⟩) × 1024 and pick the closest option.
  advanced · programming · VQE for Molecules
• Numericcorenew
  After this circuit, what is P(|01⟩) for the whole 2-qubit register measured together (qubit 0 leftmost)?
  advanced · programming · VQE for Molecules
• Multiple choicecorenew
  Classical coupled cluster writes the wavefunction as e^T applied to the Hartree-Fock state. Why can't e^T be implemented directly as a quantum circuit, and what does the Unitary Coupled-Cluster (UCC) ansatz do to fix it?
  advanced · programming · Unitary Coupled-Cluster Ansatz
• Multiple choicecorenew
  At one bond length the effective two-level molecular Hamiltonian is the matrix $\begin{bmatrix}a & c \\ c & b\end{bmatrix}$ with $a = -1.0$, $b = -0.5$, $c = 0.3$. Its lower eigenvalue is $(a+b)/2 - \sqrt{((a-b)/2)^2 + c^2}$. What ground-state energy does this point on the dissociation curve give?
  advanced · programming · Computing Dissociation Curves
• Multiple choicecorenew
  How does error mitigation (e.g. zero-noise extrapolation) differ from full quantum error correction in terms of resource overhead?
  advanced · programming · Error Mitigation in Chemistry
• Multiple choicecorenew
  In the single-excitation subspace $\{|01\rangle, |10\rangle\}$, an H2 Hamiltonian gives the 2x2 block $\begin{bmatrix}a & c \\ c & b\end{bmatrix}$ with $a = +0.2066$, $b = -1.3498$, and off-diagonal coupling $c = g_4 + g_5 = 0.0910 + 0.0910 = 0.182$. The ground-state energy is $(a+b)/2 - \sqrt{((a-b)/2)^2 + c^2}$. What is it (Hartree)?
  advanced · programming · Checkpoint: Ground-State Energy
• True / Falsecorenew
  Using the state-overlap expressibility metric of Sim, Johnson, and Aspuru-Guzik, a SMALLER Kullback-Leibler divergence between the ansatz's fidelity distribution and the Haar distribution P_Haar(F) = (d-1)(1-F)^(d-2) indicates HIGHER expressibility.
  advanced · programming · Ansatz Expressibility
• Multiple choicecorenew
  For a sufficiently random (approximately 2-design) ansatz on n qubits, how does the variance of each gradient component of the cost scale, and what does this imply for training?
  advanced · programming · Barren Plateaus
• Multiple choicecorenew
  A circuit prepares $RY(\theta)|0\rangle$, so the cost $E(\theta) = \langle Z\rangle = \cos(\theta)$. Using the parameter-shift rule, the gradient is $0.5(E(\theta + \pi/2) - E(\theta - \pi/2))$. At $\theta = 0.8$, what is $dE/d\theta$?
  advanced · programming · Gradient Measurement
• Multiple choicecorenew
  What is the quantum natural gradient update rule, and what is the matrix F that preconditions the ordinary gradient?
  advanced · programming · Natural-Gradient Optimization
• Multiple choicecorenew
  In each growth iteration of ADAPT-VQE, how does the algorithm decide which operator from the pool to append to the circuit next?
  advanced · programming · Adaptive Ansätze (ADAPT-VQE)
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |00⟩?
  advanced · programming · Measurement Grouping
• Numericcorenew
  For qubit 0 after this circuit, what is ⟨X⟩, the X-axis projection of its Bloch vector?
  advanced · programming · Measurement Grouping
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 9740), how many shots land on |00⟩? Estimate P(|00⟩) × 1024 and pick the closest option.
  advanced · programming · Measurement Grouping
• Numericcorenew
  After running this circuit, what is the probability that all 2 qubits together read |01⟩ (qubit 0 leftmost)?
  advanced · programming · Measurement Grouping
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |01⟩?
  advanced · programming · Symmetry-Preserving Ansätze
• Numericcorenew
  After running this circuit, what is the expectation value ⟨Z⟩ of qubit 0 (its Bloch Z-component)?
  advanced · programming · Symmetry-Preserving Ansätze
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 9451). Predict how many give |01⟩. Estimate P(|01⟩) × 1024 and pick the closest option.
  advanced · programming · Symmetry-Preserving Ansätze
• Numericcorenew
  For this circuit, what is the probability that measuring all 2 qubits yields exactly |10⟩ (qubit 0 leftmost)?
  advanced · programming · Symmetry-Preserving Ansätze
• Multiple choicecorenew
  Why does plain VQE find only the ground state, and what must be changed to reach excited states like E_1, E_2?
  advanced · programming · Excited-State Methods
• Multiple choicecorenew
  Why is SPSA (Simultaneous Perturbation Stochastic Approximation) a default optimizer for hardware VQE under noise?
  advanced · programming · Noise-Aware Optimization
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|10⟩)?
  advanced · programming · Checkpoint: Design a Better Ansatz
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 2511). Predict how many give |10⟩. Estimate P(|10⟩) × 1024 and pick the closest option.
  advanced · programming · Checkpoint: Design a Better Ansatz
• Numericcorenew
  After running this circuit, what is the probability that all 2 qubits together read |01⟩ (qubit 0 leftmost)?
  advanced · programming · Checkpoint: Design a Better Ansatz
• Multiple choicecorenew
  In the kernel (feature-map) view of QML, why is the data encoding described as the most consequential choice?
  advanced · programming · QML Landscape
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |01⟩?
  advanced · programming · Quantum Kernels in Depth
• Predictcorenew
  Which measurement outcome is most likely for this circuit?
  advanced · programming · Quantum Kernels in Depth
• Numericcorenew
  This circuit prepares qubit 0 in some state. What expectation value ⟨X⟩ does it have along the Bloch X-axis?
  advanced · programming · Quantum Kernels in Depth
• Predictcorenew
  On the seeded simulator (seed 6993), this circuit runs for 1024 shots. About how many come out |01⟩? Estimate P(|01⟩) × 1024 and pick the closest option.
  advanced · programming · Quantum Kernels in Depth
• Numericcorenew
  Run this circuit, then measure all 2 qubits at once. What is the probability of the joint outcome |11⟩ (qubit 0 leftmost)?
  advanced · programming · Quantum Kernels in Depth
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|1⟩)?
  advanced · programming · Variational Classifiers
• Numericcorenew
  This circuit prepares qubit 0 in some state. What expectation value ⟨X⟩ does it have along the Bloch X-axis?
  advanced · programming · Variational Classifiers
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 53). How many of the shots come out |1⟩? Estimate P(|1⟩) × 1024 and pick the closest option.
  advanced · programming · Variational Classifiers
• Multiple choicecorenew
  For a quantum circuit Born machine (QCBM) on $n$ qubits using circuit $U(\theta)$ on $|0\rangle^n$, what gives the probability of observing bitstring $x$?
  advanced · programming · Quantum Generative Models
• Multiple choicecorenew
  In a quantum GAN's minimax value function, which player optimizes in which direction, and what holds at the Nash equilibrium?
  advanced · programming · Quantum GANs (Overview)
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |0⟩?
  advanced · programming · Data Re-Uploading
• Numericcorenew
  After running this circuit, what is the expectation value ⟨Y⟩ of qubit 0 (its Bloch Y-component)?
  advanced · programming · Data Re-Uploading
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 4834). Predict how many give |0⟩. Estimate P(|0⟩) × 1024 and pick the closest option.
  advanced · programming · Data Re-Uploading
• Multiple choicecorenew
  According to Cerezo et al. (2021), how does the choice between global and local cost functions affect barren plateaus in shallow QML circuits?
  advanced · programming · Trainability and Barren Plateaus in QML
• True / Falsecorenew
  According to the lesson, the fact that a quantum kernel is classically hard to compute (hard to simulate) is by itself sufficient to guarantee that the quantum model will learn better than a classical one.
  advanced · programming · Quantum Advantage in Learning
• Multiple choicecorenew
  A QML classifier outputs real scores and predicts sign(output) as the class label in {+1, -1}. Outputs = [0.82, -0.30, 0.10, -0.65, 0.45] with true labels = [1, -1, -1, -1, 1]. Counting a prediction correct when sign matches the label, what is the accuracy?
  advanced · programming · Benchmarking QML Models
• True / Falsecorenew
  In a hybrid classical-quantum network, a parameterized quantum layer outputs measured expectation values, and because each expectation has a parameter-shift gradient, the layer is differentiable and slots into ordinary backpropagation via the chain rule.
  advanced · programming · Hybrid Architectures
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |01⟩?
  advanced · programming · Checkpoint: Train a QML Model
• Predictcorenew
  Which measurement outcome is most likely for this circuit?
  advanced · programming · Checkpoint: Train a QML Model
• Numericcorenew
  This circuit prepares qubit 0 in some state. What expectation value ⟨Z⟩ does it have along the Bloch Z-axis?
  advanced · programming · Checkpoint: Train a QML Model
• Predictcorenew
  On the seeded simulator (seed 7274), this circuit runs for 1024 shots. About how many come out |01⟩? Estimate P(|01⟩) × 1024 and pick the closest option.
  advanced · programming · Checkpoint: Train a QML Model
• Numericcorenew
  Run this circuit, then measure all 2 qubits at once. What is the probability of the joint outcome |10⟩ (qubit 0 leftmost)?
  advanced · programming · Checkpoint: Train a QML Model
• Multiple choicecorenew
  In the standard sufficient condition for adiabatic evolution, the required total runtime T scales with the minimum spectral gap (the energy difference E_1 - E_0 at its smallest, written delta_min) as which of the following?
  advanced · programming · The Adiabatic Theorem
• Multiple choicecorenew
  In adiabatic quantum computation with the linear schedule H(s) = (1-s)H_B + s H_P, what is the canonical driver Hamiltonian H_B and the state the system is prepared in at s = 0?
  advanced · programming · Adiabatic Quantum Computation
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |00⟩?
  advanced · programming · Problem and Driver Hamiltonians
• Numericcorenew
  After running this circuit, what is the expectation value ⟨X⟩ of qubit 0 (its Bloch X-component)?
  advanced · programming · Problem and Driver Hamiltonians
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 5692). Predict how many give |00⟩. Estimate P(|00⟩) × 1024 and pick the closest option.
  advanced · programming · Problem and Driver Hamiltonians
• Numericcorenew
  For this circuit, what is the probability that measuring all 2 qubits yields exactly |01⟩ (qubit 0 leftmost)?
  advanced · programming · Problem and Driver Hamiltonians
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|0⟩)?
  advanced · programming · Annealing Schedules
• Numericcorenew
  This circuit prepares qubit 0 in some state. What expectation value ⟨X⟩ does it have along the Bloch X-axis?
  advanced · programming · Annealing Schedules
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 2248). How many of the shots come out |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  advanced · programming · Annealing Schedules
• Multiple choicecorenew
  Along an adiabatic schedule $s \in [0, 1]$, the instantaneous spectral gap is $\delta(s) = 2\sqrt{(1 - s)^2 + s^2}$. The minimum gap occurs at $s = 1/2$. What is the minimum gap value?
  advanced · programming · The Minimum Gap
• Multiple choicecorenew
  A two-spin Ising model has energy E = -s0*s1 with each spin si in {+1, -1}. Minimizing over all four configurations, what is the ground-state (minimum) energy?
  advanced · programming · Ising and QUBO Formulations
• True / Falsecorenew
  A landmark result establishes that adiabatic quantum computation is polynomially equivalent to the standard circuit model, which means AQC is universal for quantum computation rather than merely an optimization heuristic.
  advanced · programming · Equivalence to the Circuit Model
• Multiple choicecorenew
  The Landau-Zener formula gives the diabatic transition probability near an avoided crossing as $P_\text{diabatic} = \exp(-\pi \Delta_\text{min}^2 / (2 \hbar \alpha))$. How do the minimum gap $\Delta_\text{min}$ and the sweep rate $\alpha$ affect the chance of an error (jumping out of the ground state)?
  advanced · programming · Diabatic Transitions
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|0⟩)?
  advanced · programming · Simulating a Small Anneal
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |0⟩?
  advanced · programming · Checkpoint: Anneal to a Ground State
• Multiple choicecorenew
  A general state of n qubits lives in a Hilbert space of dimension 2^n. Storing a general state by brute force as a full state vector therefore requires how many complex amplitudes?
  advanced · programming · Why Tensor Networks
• Multiple choicecorenew
  When building a matrix product state, the singular values produced at each cut by the iterated SVD are exactly which quantity for splitting the chain at that bond?
  advanced · programming · Matrix Product States
• Numericcorenew
  After running this circuit, what is the probability of measuring |000⟩?
  advanced · programming · Bond Dimension and Entanglement
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 2912), how many shots land on |000⟩? Estimate P(|000⟩) × 1024 and pick the closest option.
  advanced · programming · Bond Dimension and Entanglement
• Numericcorenew
  For this circuit, what is the probability that measuring all 3 qubits yields exactly |111⟩ (qubit 0 leftmost)?
  advanced · programming · Bond Dimension and Entanglement
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |1⟩?
  advanced · programming · Contracting Tensor Networks
• True / Falsecorenew
  When simulating a circuit with an MPS, applying a single-qubit gate cannot create entanglement and leaves the bond dimensions untouched, whereas a two-qubit gate on neighboring sites can grow the bond dimension (it can double the local bond), which is then controlled by an SVD truncation.
  advanced · programming · Simulating Circuits with MPS
• Multiple choicecorenew
  A quantum state obeys an 'area law' for entanglement entropy when, for a cut-out region A, the entropy S(A) scales with which feature of A?
  advanced · programming · Area Laws
• Multiple choicecorenew
  DMRG finds ground states by exploiting the variational principle. When it optimizes a single site tensor with all neighboring tensors held fixed, minimizing the energy reduces to which kind of standard problem?
  advanced · programming · DMRG (Overview)
• Multiple choicecorenew
  A PEPS stores a 2D state with only O(n d D^4) parameters (polynomial in the number of sites). According to the lesson, what is the hard part for a general PEPS?
  advanced · programming · PEPS and 2D Systems
• True / Falsecorenew
  According to the lesson, the GHZ state is a counterexample to the idea that 'highly entangled always means classically hard': it is highly correlated yet has bond dimension chi = 2, because what kills tensor networks is high entanglement entropy across a cut (many comparable Schmidt coefficients), not correlation per se.
  advanced · programming · Limits of Classical Simulation
• Multiple choicecorenew
  Across a bipartition, a state's squared Schmidt coefficients are [0.5, 0.5, 0, 0]. The Schmidt rank (bond dimension) chi is the count of nonzero coefficients (above tolerance). What is chi?
  advanced · programming · Checkpoint: Estimate Simulability
• Multiple choicecorenew
  In the quantum metrology pipeline, the unknown parameter $\theta$ is imprinted on the probe during the 'encoding' stage through which kind of operation?
  advanced · programming · The Metrology Problem
• Numericcorenew
  After running this circuit, what is the probability of measuring |0⟩?
  advanced · programming · The Standard Quantum Limit
• Numericcorenew
  Run this circuit. What is qubit 0's Bloch Y-component — the expectation value ⟨Y⟩?
  advanced · programming · The Standard Quantum Limit
• Predictcorenew
  On the seeded simulator (seed 1719), this circuit runs for 1024 shots. About how many come out |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  advanced · programming · The Standard Quantum Limit
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |0⟩?
  advanced · programming · Quantum Fisher Information
• Numericcorenew
  After running this circuit, what is the expectation value ⟨X⟩ of qubit 0 (its Bloch X-component)?
  advanced · programming · Quantum Fisher Information
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 4752). Predict how many give |0⟩. Estimate P(|0⟩) × 1024 and pick the closest option.
  advanced · programming · Quantum Fisher Information
• Multiple choicecorenew
  For $M$ independent measurements of a parameter with quantum Fisher information $F_Q$ per shot, the Cramér–Rao bound on the standard deviation is $1 / \sqrt{M F_Q}$. With $M = 100$ and $F_Q = 1$, what is the smallest achievable standard deviation?
  advanced · programming · The Cramér–Rao Bound
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |0⟩?
  advanced · programming · Ramsey Interferometry
• Numericcorenew
  For qubit 0 after this circuit, what is ⟨Y⟩, the Y-axis projection of its Bloch vector?
  advanced · programming · Ramsey Interferometry
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 8659), how many shots land on |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  advanced · programming · Ramsey Interferometry
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |000⟩?
  advanced · programming · Entanglement-Enhanced Sensing
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 8370). How many of the shots come out |000⟩? Estimate P(|000⟩) × 1024 and pick the closest option.
  advanced · programming · Entanglement-Enhanced Sensing
• Numericcorenew
  After this circuit, what is P(|111⟩) for the whole 3-qubit register measured together (qubit 0 leftmost)?
  advanced · programming · Entanglement-Enhanced Sensing
• Numericcorenew
  After running this circuit, what is the probability of measuring |00⟩?
  advanced · programming · The Heisenberg Limit
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 5337), how many shots land on |00⟩? Estimate P(|00⟩) × 1024 and pick the closest option.
  advanced · programming · The Heisenberg Limit
• Numericcorenew
  For this circuit, what is the probability that measuring all 2 qubits yields exactly |11⟩ (qubit 0 leftmost)?
  advanced · programming · The Heisenberg Limit
• Multiple choicecorenew
  Under independent (uncorrelated Markovian) dephasing, the Huelga result shows the best entangled probe strategy achieves what scaling of the phase uncertainty with the number of probes N?
  advanced · programming · Noise in Metrology
• Multiple choicecorenew
  The lesson describes an atomic clock as a continuously-run version of which interferometric sensing protocol from the module?
  advanced · programming · Applications: Clocks and Magnetometers
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |00⟩?
  advanced · programming · Checkpoint: Beat the Standard Limit
• Numericcorenew
  After running this circuit, what is the expectation value ⟨Y⟩ of qubit 0 (its Bloch Y-component)?
  advanced · programming · Checkpoint: Beat the Standard Limit
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 556). Predict how many give |00⟩. Estimate P(|00⟩) × 1024 and pick the closest option.
  advanced · programming · Checkpoint: Beat the Standard Limit
• Numericcorenew
  For this circuit, what is the probability that measuring all 2 qubits yields exactly |10⟩ (qubit 0 leftmost)?
  advanced · programming · Checkpoint: Beat the Standard Limit
• Multiple choicecorenew
  The lesson says a classical signal lost in fibre can be re-amplified by repeaters, but an unknown qubit cannot be amplified the same way. Which principle forbids this amplification?
  advanced · programming · The Quantum Internet Vision
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|00⟩)?
  advanced · programming · Entanglement Distribution
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 97). Predict how many give |00⟩. Estimate P(|00⟩) × 1024 and pick the closest option.
  advanced · programming · Entanglement Distribution
• Numericcorenew
  After running this circuit, what is the probability that all 2 qubits together read |11⟩ (qubit 0 leftmost)?
  advanced · programming · Entanglement Distribution
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |0000⟩?
  advanced · programming · Entanglement Swapping
• Predictcorenew
  On the seeded simulator (seed 3130), this circuit runs for 1024 shots. About how many come out |0000⟩? Estimate P(|0000⟩) × 1024 and pick the closest option.
  advanced · programming · Entanglement Swapping
• Numericcorenew
  Run this circuit, then measure all 4 qubits at once. What is the probability of the joint outcome |0010⟩ (qubit 0 leftmost)?
  advanced · programming · Entanglement Swapping
• Multiple choicecorenew
  Distributing a Bell pair directly over fibre has cost growing exponentially with distance L. By splitting the span into short elementary links and using nested entanglement swapping, a quantum repeater changes the cost to grow how with L?
  advanced · programming · Quantum Repeaters
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |1001⟩?
  advanced · programming · Distributed Gates via Teleportation
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 9196). How many of the shots come out |1001⟩? Estimate P(|1001⟩) × 1024 and pick the closest option.
  advanced · programming · Distributed Gates via Teleportation
• Numericcorenew
  After this circuit, what is P(|1011⟩) for the whole 4-qubit register measured together (qubit 0 leftmost)?
  advanced · programming · Distributed Gates via Teleportation
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |0000⟩?
  advanced · programming · Entanglement Purification
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 9047). How many of the shots come out |0000⟩? Estimate P(|0000⟩) × 1024 and pick the closest option.
  advanced · programming · Entanglement Purification
• Numericcorenew
  After this circuit, what is P(|0011⟩) for the whole 4-qubit register measured together (qubit 0 leftmost)?
  advanced · programming · Entanglement Purification
• Multiple choicecorenew
  In the proposed quantum network protocol stack, what is the primary job of the link layer?
  advanced · programming · Quantum Network Protocols
• True / Falsecorenew
  In the BFK blind quantum computing protocol, the client must possess a full quantum computer in order to keep its algorithm and data private from the server.
  advanced · programming · Blind Quantum Computing
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|000⟩)?
  advanced · programming · Multiparty Entanglement
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 2107). Predict how many give |000⟩. Estimate P(|000⟩) × 1024 and pick the closest option.
  advanced · programming · Multiparty Entanglement
• Numericcorenew
  After running this circuit, what is the probability that all 3 qubits together read |111⟩ (qubit 0 leftmost)?
  advanced · programming · Multiparty Entanglement
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |0000⟩?
  advanced · programming · Checkpoint: A Two-Node Protocol
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 1233). How many of the shots come out |0000⟩? Estimate P(|0000⟩) × 1024 and pick the closest option.
  advanced · programming · Checkpoint: A Two-Node Protocol
• Numericcorenew
  After this circuit, what is P(|0010⟩) for the whole 4-qubit register measured together (qubit 0 leftmost)?
  advanced · programming · Checkpoint: A Two-Node Protocol
• True / Falsecorenew
  In benchmarking a quantum device, the error rate (infidelity) r is defined as r = 1 - F, where F is the gate fidelity. True or false?
  advanced · programming · Why Benchmark
• Numericcorenew
  After running this circuit, what is the probability of measuring |0⟩?
  advanced · programming · Quantum State Tomography
• Multiple choicecorenew
  For single-qubit quantum process tomography, which set of input states is sufficient to span the space of density matrices so that the channel can be fully reconstructed?
  advanced · programming · Quantum Process Tomography
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|0⟩)?
  advanced · programming · Randomized Benchmarking
• Multiple choicecorenew
  In interleaved randomized benchmarking (IRB), the interleaved sequence is built so that the extra decay it shows (relative to a reference RB run) can be attributed to the target gate G. How is the interleaved sequence constructed?
  advanced · programming · Interleaved RB
• Multiple choicecorenew
  In cross-entropy benchmarking (XEB), the linear cross-entropy fidelity is F_XEB = 2^n * <p_ideal(x)> - 1, the rescaled average ideal probability of the device's observed bitstrings. What value does F_XEB approach for a fully noisy device that samples uniformly?
  advanced · programming · Cross-Entropy Benchmarking
• Numericcorenew
  After running this circuit, what is the probability of measuring |00⟩?
  advanced · programming · Quantum Volume
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 1039), how many shots land on |00⟩? Estimate P(|00⟩) × 1024 and pick the closest option.
  advanced · programming · Quantum Volume
• Numericcorenew
  For this circuit, what is the probability that measuring all 2 qubits yields exactly |01⟩ (qubit 0 leftmost)?
  advanced · programming · Quantum Volume
• Multiple choicecorenew
  A $2\times 2$ readout confusion matrix is $A = \begin{bmatrix}a & b \\ c & d\end{bmatrix} = \begin{bmatrix}0.90 & 0.05 \\ 0.10 & 0.95\end{bmatrix}$, mapping true probabilities to measured ones. Measured probabilities are $(p_\text{Meas0}, p_\text{Meas1}) = (0.4, 0.6)$. Inverting ($\det = ad - bc$), the corrected $p_\text{True0} = (d\,p_\text{Meas0} - b\,p_\text{Meas1})/\det$. What is $p_\text{True0}$?
  advanced · programming · Readout Error Mitigation
• Multiple choicecorenew
  Zero-noise extrapolation: an expectation value is measured at two noise scale factors. At $\lambda=1$ the result is $E=0.7$, and at $\lambda=3$ it is $E=0.5$. Fitting a straight line $E(\lambda)=a\lambda+E_0$ and extrapolating to the zero-noise limit ($\lambda=0$), what is the mitigated value $E_0$?
  advanced · programming · Zero-Noise Extrapolation
• Multiple choicecorenew
  In single-qubit (d = 2) randomized benchmarking, the survival decays as F(m) = A*p^m + B with B = 1/d = 0.5. Measured points: F(1) = 0.99 and F(5) = 0.95196. From p = ((F(5)-B)/(F(1)-B))^(1/(5-1)), the average gate fidelity is F = 1 - (d-1)(1-p)/d. What is F?
  advanced · programming · Checkpoint: Estimate Gate Fidelity
• Multiple choicecorenew
  In the quantum software stack, where does a state-vector simulator like this platform's fit, and what is it good (and not good) for?
  advanced · programming · The Quantum Software Stack
• Multiple choicecorenew
  OpenQASM and QIR are the two dominant quantum intermediate representations. Which statement correctly contrasts them?
  advanced · programming · Intermediate Representations (QIR/QASM)
• Multiple choicecorenew
  For Grover search over $N=2^{10}=1024$ items, the optimal number of iterations is $k=\lfloor(\pi/4)\sqrt{N}\rfloor$. What is $k$?
  advanced · programming · Resource Estimation
• Multiple choicecorenew
  A fault-tolerant circuit contains 12 Toffoli gates plus 4 standalone T-gates. Each Toffoli decomposes into 7 T-gates. What is the total T-count of the circuit?
  advanced · programming · Counting T-Gates and Qubits
• Multiple choicecorenew
  A surface-code computation needs 50 logical qubits at code distance $d=15$. Using the estimate of $2d^2$ physical qubits per logical qubit, how many physical qubits are required in total?
  advanced · programming · Logical vs Physical Resource Costs
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |00⟩?
  advanced · programming · Testing Quantum Programs
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 9156). How many of the shots come out |00⟩? Estimate P(|00⟩) × 1024 and pick the closest option.
  advanced · programming · Testing Quantum Programs
• Numericcorenew
  After this circuit, what is P(|11⟩) for the whole 2-qubit register measured together (qubit 0 leftmost)?
  advanced · programming · Testing Quantum Programs
• Multiple choicecorenew
  Why are phase bugs called 'the silent killer' when debugging quantum circuits, and where can you reliably catch them?
  advanced · programming · Debugging with Simulators
• True / Falsecorenew
  When version-controlling quantum circuits, the lesson's first rule is to commit the highest-level description (the generator code and its fixed parameters) and to treat the transpiled circuit and measurement results as derived outputs. True or false?
  advanced · programming · Version Control for Circuits
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|00⟩)?
  advanced · programming · Reproducibility and Seeds
• Numericcorenew
  This circuit prepares qubit 0 in some state. What expectation value ⟨X⟩ does it have along the Bloch X-axis?
  advanced · programming · Reproducibility and Seeds
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 2216). How many of the shots come out |00⟩? Estimate P(|00⟩) × 1024 and pick the closest option.
  advanced · programming · Reproducibility and Seeds
• Numericcorenew
  Run this circuit, then measure all 2 qubits at once. What is the probability of the joint outcome |10⟩ (qubit 0 leftmost)?
  advanced · programming · Reproducibility and Seeds
• Multiple choicecorenew
  Resource estimate for Grover search on $n=8$ qubits ($N=2^8=256$ items). The optimal iteration count is $k=\lfloor(\pi/4)\sqrt{N}\rfloor$. Each Grover iterate uses 5 Toffoli gates and each Toffoli costs 7 T-gates. What is the total T-gate count $k \cdot 5 \cdot 7$?
  advanced · programming · Checkpoint: Estimate Algorithm Resources
• Multiple choicecorenew
  In Shor's algorithm to factor $N=15$ with base $a=7$, the order-finding step yields period $r=4$. The factors are obtained from $\gcd(a^{r/2} - 1, N)$ and $\gcd(a^{r/2} + 1, N)$. Since $7^2 = 49 = 4 \pmod{15}$, what nontrivial factor does $\min(\gcd(4-1,15), \gcd(4+1,15))$ return?
  advanced · programming · Project: End-to-End Shor
• Numericcorenew
  After running this circuit, what is the probability of measuring |00011⟩?
  advanced · programming · Project: A Full QEC Demo
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 7603), how many shots land on |00011⟩? Estimate P(|00011⟩) × 1024 and pick the closest option.
  advanced · programming · Project: A Full QEC Demo
• Numericcorenew
  For this circuit, what is the probability that measuring all 5 qubits yields exactly |11111⟩ (qubit 0 leftmost)?
  advanced · programming · Project: A Full QEC Demo
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |01⟩?
  advanced · programming · Project: Molecular Ground State
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 663). How many of the shots come out |01⟩? Estimate P(|01⟩) × 1024 and pick the closest option.
  advanced · programming · Project: Molecular Ground State
• Numericcorenew
  After this circuit, what is P(|10⟩) for the whole 2-qubit register measured together (qubit 0 leftmost)?
  advanced · programming · Project: Molecular Ground State
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|000⟩)?
  advanced · programming · Project: QAOA at Scale
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 1537). Predict how many give |000⟩. Estimate P(|000⟩) × 1024 and pick the closest option.
  advanced · programming · Project: QAOA at Scale
• Numericcorenew
  After running this circuit, what is the probability that all 3 qubits together read |111⟩ (qubit 0 leftmost)?
  advanced · programming · Project: QAOA at Scale
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |1⟩?
  advanced · programming · Project: A Quantum Classifier
• Multiple choicecorenew
  Error-mitigated estimation via linear zero-noise extrapolation: at noise scale s=1 the measured expectation is 0.80, and at s=2 it is 0.64. Fitting E(s)=b*s+a and reading off the zero-noise intercept a, what is the mitigated estimate?
  advanced · programming · Project: Error-Mitigated Estimation
• Numericcorenew
  After running this circuit, what is the probability of measuring |0000⟩?
  advanced · programming · Project: Entanglement Network
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 1248), how many shots land on |0000⟩? Estimate P(|0000⟩) × 1024 and pick the closest option.
  advanced · programming · Project: Entanglement Network
• Numericcorenew
  For this circuit, what is the probability that measuring all 4 qubits yields exactly |0011⟩ (qubit 0 leftmost)?
  advanced · programming · Project: Entanglement Network
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |0⟩?
  advanced · programming · Project: Benchmark a Device Model
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|00⟩)?
  advanced · programming · Project: Hamiltonian Simulation Study
• Predictcorenew
  After running this circuit, which outcome has the highest probability?
  advanced · programming · Project: Hamiltonian Simulation Study
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 7314). How many of the shots come out |00⟩? Estimate P(|00⟩) × 1024 and pick the closest option.
  advanced · programming · Project: Hamiltonian Simulation Study
• Multiple choicecorenew
  Resource estimate for a real algorithm requiring 100 logical qubits at surface-code distance $d=15$. With $2d^2$ physical qubits per logical qubit, how many physical qubits are needed in total?
  advanced · programming · Project: Resource Estimate a Real Algorithm
• Multiple choicecorenew
  How does the lesson characterize the current 'NISQ era' of quantum hardware?
  advanced · programming · Reflection: The Path Forward
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |0000⟩?
  advanced · programming · Capstone: Original Research Demo
• Predictcorenew
  On the seeded simulator (seed 2533), this circuit runs for 1024 shots. About how many come out |0000⟩? Estimate P(|0000⟩) × 1024 and pick the closest option.
  advanced · programming · Capstone: Original Research Demo
• Numericcorenew
  Run this circuit, then measure all 4 qubits at once. What is the probability of the joint outcome |1111⟩ (qubit 0 leftmost)?
  advanced · programming · Capstone: Original Research Demo
• Multiple choicecorenew
  In quantum mechanics, the Born rule converts a complex probability amplitude $\psi(x)$ into the probability of outcome $x$. What is that probability?
  beginner · physics · Why Complex Numbers in Quantum Theory
• Multiple choicecorenew
  The powers of i cycle with period 4: i¹ = i, i² = −1, i³ = −i, i⁴ = 1. What is i²²?
  beginner · physics · The Imaginary Unit i
• Multiple choicecorenew
  What is the real part of the product (2 + 3i)(1 − 2i)?
  beginner · physics · Complex Arithmetic
• Multiple choicecorenew
  On the Argand diagram, what is the geometric effect of multiplying a complex number $z$ by $e^{i\theta}$?
  beginner · physics · The Complex Plane
• Multiple choicecorenew
  What is the modulus |3 + 4i|?
  beginner · physics · Modulus and Argument
• Multiple choicecorenew
  For z = 5 − 12i, the product z·z̄ equals |z|². What is |z| = √(z·z̄)?
  beginner · physics · Complex Conjugate
• Multiple choicecorenew
  Written in polar form r·e^{iθ}, what is the modulus r of z = 1 + √3·i?
  beginner · physics · Polar Form
• Multiple choicecorenew
  By Euler’s formula e^{iθ} = cos θ + i sin θ, what is the real part of e^{iπ/3}?
  beginner · physics · Euler’s Formula
• Multiple choicecorenew
  The primitive cube roots of unity are e^{±2πi/3}. What is the real part of e^{2πi/3}?
  beginner · physics · Roots of Unity
• Multiple choicecorenew
  A phase e^{iφ} is a unit complex number (modulus 1). For φ = π/3, what is its real part?
  beginner · physics · Phases as Unit Complex Numbers
• Multiple choicecorenew
  Evaluate |3 + 4i| + Re((2 + i)²) + Re(e^{iπ/3}) + |e^{2πi/5}|.
  beginner · physics · Checkpoint: Complex Number Workout
• Multiple choicecorenew
  For the qubit state (1/√2)|0⟩ + (1/√2)|1⟩, what is the normalization sum |α|² + |β|²?
  beginner · physics · Vectors and Column Notation
• Multiple choicecorenew
  Multiply the complex number (1 + 2i) by i, then add (2 − i). What is the modulus of the result?
  beginner · physics · Vector Addition and Scaling
• Multiple choicecorenew
  For u = (1 + i, 2) and v = (3, 1 − i), the Hermitian inner product is ⟨u|v⟩ = 5 − 5i. What is its magnitude |⟨u|v⟩|?
  beginner · physics · The Inner Product
• Multiple choicecorenew
  The vector v = (1, 2, 2) has norm ‖v‖ = 3. After normalizing (v/‖v‖), what is its first component?
  beginner · physics · Norm and Normalization
• Multiple choicecorenew
  What is the inner product (3, 4)·(4, −3), and what does it tell you about the two vectors?
  beginner · physics · Orthogonality
• Multiple choicecorenew
  Applying the matrix $M = \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}$ to $v = (1, 0)^\top$, what is the first component of $Mv$?
  beginner · physics · Matrices as Linear Maps
• Multiple choicecorenew
  For $A = \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}$ (the X gate) and $B = \begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}$ (the Z gate), what is the $(0, 1)$ entry of the product $AB$?
  beginner · physics · Matrix Multiplication
• Multiple choicecorenew
  For $M = \begin{bmatrix}1 + 2i & 3 \\ 0 & -1 + i\end{bmatrix}$, the adjoint $M^\dagger$ is the conjugate transpose. What is the $(0, 1)$ entry of $M^\dagger$?
  beginner · physics · The Adjoint (Dagger)
• True / Falsecorenew
  A matrix is Hermitian if it equals its own conjugate transpose ($M^\dagger = M$). Is $M = \begin{bmatrix}3 & 1 + 2i \\ 1 - 2i & 5\end{bmatrix}$ Hermitian?
  beginner · physics · Hermitian Matrices
• Multiple choicecorenew
  The Hadamard matrix has first column (1/√2, 1/√2). What is the (0, 0) entry of H†H (which equals the identity for a unitary)?
  beginner · physics · Unitary Matrices
• Multiple choicecorenew
  The matrix $A = \begin{bmatrix}3 & 1 \\ 1 & 3\end{bmatrix}$ has characteristic equation $(3 - \lambda)^2 - 1 = 0$. What is its larger eigenvalue?
  beginner · physics · Eigenvalues and Eigenvectors
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|0⟩)?
  beginner · physics · Checkpoint: Linear Algebra for Qubits
• Numericcorenew
  This circuit prepares qubit 0 in some state. What expectation value ⟨X⟩ does it have along the Bloch X-axis?
  beginner · physics · Checkpoint: Linear Algebra for Qubits
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 7738). How many of the shots come out |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  beginner · physics · Checkpoint: Linear Algebra for Qubits
• Multiple choicecorenew
  For a normalized wavefunction, the definite integral $\int_a^b |\psi(x)|^2\,dx$ computes which quantity?
  beginner · physics · Calculus Essentials: Definite Integrals
• Multiple choicecorenew
  Applying the power rule term by term, which is an antiderivative of $f(x) = x^2 - 2x^3$?
  beginner · physics · Calculus Essentials: Definite Integrals
• True / Falsecorenew
  By the Fundamental Theorem of Calculus, the definite integral $\int_a^b f(x)\,dx$ equals $F(a) - F(b)$, where $F$ is any antiderivative of $f$.
  beginner · physics · Calculus Essentials: Definite Integrals
• Multiple choicecorenew
  In the lesson's vocabulary, what is an 'event' for an experiment with sample space Omega?
  beginner · physics · Sample Spaces and Events
• Multiple choicecorenew
  Rolling a die, A = {1, 2, 3} and B = {3, 4, 5}. By inclusion–exclusion, what is P(A ∪ B)?
  beginner · physics · Probability Axioms
• Multiple choicecorenew
  In the lesson's definition, what is a random variable X on a sample space Omega?
  beginner · physics · Random Variables
• Multiple choicecorenew
  What is the expectation value (mean) of a single roll of a fair six-sided die?
  beginner · physics · Expectation Value
• Multiple choicecorenew
  A measurement gives +1 with probability 3/4 and −1 with probability 1/4. What is the standard deviation σ?
  beginner · physics · Variance and Standard Deviation
• Multiple choicecorenew
  A qubit is in state α|0⟩ + β|1⟩ with α = √3/2. What is the probability of measuring |0⟩?
  beginner · physics · Distributions of Measurement Outcomes
• Multiple choicecorenew
  According to the lesson, which statement about the relationship between independence and correlation is correct?
  beginner · physics · Independence and Correlation
• Multiple choicecorenew
  A coin lands heads 520 times in 1000 flips. What is the observed relative frequency of heads?
  beginner · physics · The Law of Large Numbers
• Multiple choicecorenew
  An estimate p̂ = 0.4 comes from n = 100 samples. What is the standard error √(p̂(1 − p̂)/n)?
  beginner · physics · Sampling Error
• Multiple choicecorenew
  A qubit measures 0 with probability 1/3 and 1 with probability 2/3 (outcome X = 0 or 1). What is the variance of X?
  beginner · physics · Checkpoint: Statistics of a Qubit
• Multiple choicecorenew
  For a mass m on a spring with spring constant k undergoing simple harmonic motion, what is the angular frequency omega?
  beginner · physics · Oscillations and Frequency
• Multiple choicecorenew
  An electromagnetic wave travels at $c = 3.0\times 10^{8}$ m/s with frequency $f = 100$ MHz $= 100\times 10^{6}$ Hz. Using $\lambda = c/f$, what is its wavelength in meters?
  beginner · physics · Wave Basics: Wavelength and Period
• Multiple choicecorenew
  Two waves with amplitudes A1 = 3.0 and A2 = 4.0 meet exactly in phase (fully constructive interference), so their amplitudes add directly. What is the resulting amplitude?
  beginner · physics · Superposition of Waves
• Multiple choicecorenew
  In the classical double-slit experiment with slit separation $d$ and wavelength $\lambda$, a bright fringe (constructive interference) occurs at angle $\theta$ when:
  beginner · physics · Interference and the Double Slit
• Multiple choicecorenew
  For single-slit diffraction of a plane wave (wavelength $\lambda$) through a slit of width $a$, the condition $a\sin\theta = m\lambda$ for $m = \pm1, \pm2, \ldots$ gives the angles of which features?
  beginner · physics · Diffraction
• Multiple choicecorenew
  According to the lesson, in a monochromatic electromagnetic plane wave in vacuum, how do the electric field E and magnetic field B oscillate relative to the propagation direction?
  beginner · physics · Electromagnetic Waves
• Multiple choicecorenew
  The energy carried by a classical wave is proportional to the square of its amplitude. Wave 2 has amplitude A2 = 6 and wave 1 has amplitude A1 = 3. What is the ratio of their energies E2/E1?
  beginner · physics · Energy in Classical Waves
• True / Falsecorenew
  In the photoelectric effect, below a critical threshold frequency nu_0 no electrons are emitted at all, no matter how intense the light.
  beginner · physics · The Photoelectric Puzzle (Setup)
• Multiple choicecorenew
  Why does excited hydrogen gas produce a discrete set of sharp spectral lines rather than a continuous rainbow?
  beginner · physics · Spectra and Discrete Lines
• Multiple choicecorenew
  In a double-slit experiment, light of wavelength $\lambda = 500$ nm illuminates slits separated by $d = 0.25$ mm. On a screen at distance $L = 1.0$ m, the fringe spacing is $\Delta y = \lambda L/d$. What is $\Delta y$ in millimeters?
  beginner · physics · Checkpoint: Classical Waves Recap
• Multiple choicecorenew
  What assumption did Planck introduce to derive his blackbody formula and eliminate the ultraviolet catastrophe?
  beginner · physics · Blackbody Radiation
• Multiple choicecorenew
  A photon has frequency nu = 6.0e14 Hz. Its energy is E = h*nu with h = 6.626e-34 J*s. Converting to electron-volts (1 eV = 1.602e-19 J), what is the photon energy in eV?
  beginner · physics · Planck’s Quantum Hypothesis
• Multiple choicecorenew
  In the photoelectric effect, the maximum kinetic energy of an ejected electron is KE_max = (photon energy) - (work function). A photon of energy 3.0 eV strikes a metal with work function 2.0 eV. What is KE_max in eV?
  beginner · physics · The Photoelectric Effect
• Multiple choicecorenew
  A monochromatic light beam of frequency nu is made of photons. What is the energy carried by a single photon?
  beginner · physics · Photons
• Multiple choicecorenew
  In Compton scattering the wavelength shift is $\Delta\lambda = \lambda_C(1 - \cos\theta)$, where $\lambda_C = 2.426\times 10^{-12}$ m is the electron Compton wavelength. For scattering at $\theta = 90$ degrees, what is $\Delta\lambda$ in meters?
  beginner · physics · The Compton Effect
• Multiple choicecorenew
  Using the Rydberg formula $1/\lambda = R_H(1/n_1^2 - 1/n_2^2)$ with $R_H = 1.097\times 10^{7} \text{ m}^{-1}$, find the wavelength of the hydrogen transition from $n_2 = 3$ to $n_1 = 2$ (the H-alpha line). What is $\lambda$ in nm?
  beginner · physics · Atomic Spectra and the Rydberg Formula
• Multiple choicecorenew
  In the Bohr model of hydrogen the energy levels are $E_n = -13.6/n^2$ eV. What is the energy of the $n = 2$ level in eV?
  beginner · physics · The Bohr Model
• Multiple choicecorenew
  The lesson notes the Bohr model gets hydrogen's energy levels right but is wrong about the ground state's angular momentum. What does full quantum mechanics give for the angular momentum of the hydrogen ground state?
  beginner · physics · Successes and Failures of Bohr
• Multiple choicecorenew
  An electron ($m = 9.109\times 10^{-31}$ kg) is accelerated through $V = 100$ V, gaining kinetic energy $K = eV$ with $e = 1.602\times 10^{-19}$ C. Its momentum is $p = \sqrt{2mK}$ and de Broglie wavelength $\lambda = h/p$ with $h = 6.626\times 10^{-34} \text{ J·s}$. What is $\lambda$ in meters?
  beginner · physics · de Broglie’s Matter Waves
• Multiple choicecorenew
  According to the lesson, which fundamental physical constant appears in every milestone of the old quantum theory (Planck 1900, Einstein 1905, Bohr 1913, Compton 1923, de Broglie 1924)?
  beginner · physics · Old Quantum Theory in Context
• Multiple choicecorenew
  In hydrogen the electron drops from level n=3 (E3 = -13.6/9 eV) to level n=2 (E2 = -13.6/4 eV). The emitted photon's energy equals E3 - E2. What is that photon energy in eV?
  beginner · physics · Checkpoint: Quanta and Photons
• Multiple choicecorenew
  The de Broglie relation assigns a wavelength to a particle of momentum p. What is that wavelength (h is Planck's constant)?
  beginner · physics · Duality, Stated Carefully
• True / Falsecorenew
  In the Davisson-Germer experiment (1927), a beam of electrons scattered off a nickel crystal produced a sharp intensity peak at a specific angle, consistent with waves diffracting off the regular atomic planes via the Bragg condition $2 d \sin\theta = n\lambda$. True or false: this sharp diffraction peak is exactly what a beam of classical particles would also produce.
  beginner · physics · Electron Diffraction
• Multiple choicecorenew
  In the single-particle double-slit experiment, the probability of detecting a particle at a given position on the screen (both slits open) is given by $|\psi_1 + \psi_2|^2$, where $\psi_1$ and $\psi_2$ are the probability amplitudes for the two paths. What does this differ from, and why does it produce interference?
  beginner · physics · The Double Slit with Single Particles
• Multiple choicecorenew
  An electron ($m = 9.109e-31$ kg) is accelerated through $V = 54$ V (the Davisson-Germer energy), so $p = \sqrt{2 m e V}$ with $e = 1.602e-19$ C. Its de Broglie wavelength is $\lambda = h/p$ with $h = 6.626e-34$ J*s. What is $\lambda$ in picometers ($1$ pm $= 1e-12$ m)?
  beginner · physics · Matter Wavelength Calculations
• Multiple choicecorenew
  A Gaussian wave packet has spatial width sigma_x and momentum width sigma_p. What value does the product sigma_x * sigma_p take for this packet, and what is its significance?
  beginner · physics · Wave Packets
• Multiple choicecorenew
  For a free non-relativistic electron (m = 9.109e-31 kg, hbar = 1.0546e-34 J*s), the group velocity is v_g = hbar*k/m. For wavenumber k = 1.0e10 m^-1, what is v_g in m/s?
  beginner · physics · Group and Phase Velocity
• Multiple choicecorenew
  Bohr's principle of complementarity, as applied to the double-slit experiment, states that:
  beginner · physics · Complementarity
• True / Falsecorenew
  In the probability-amplitude (Born) interpretation, the wavefunction $\Psi$ is complex-valued and the measurable probability density is $|\Psi|^2$. True or false: because $|\Psi|^2$ discards the phase of $\Psi$, the phase is physically irrelevant and plays no role in any observable.
  beginner · physics · Probability Waves
• Multiple choicecorenew
  According to the Born rule as stated for a 1D wavefunction $\psi(x,t)$, what is the probability of finding the particle in the interval $[a, b]$ at time $t$?
  beginner · physics · From Duality to the Wavefunction
• Multiple choicecorenew
  An electron ($m = 9.109\times 10^{-31}$ kg) is accelerated through 150 V, so $KE = 150e$ with $e = 1.602\times 10^{-19}$ C. Its momentum is $p = \sqrt{2m \cdot KE}$ and de Broglie wavelength $\lambda = h/p$ with $h = 6.626\times 10^{-34} \text{ J·s}$. What is $\lambda$ in meters?
  beginner · physics · Checkpoint: Matter Waves
• Multiple choicecorenew
  For a single particle in one dimension, what mathematical condition expresses that the particle must be found somewhere (the normalization of the wavefunction $\psi$)?
  beginner · physics · The Wavefunction ψ(x)
• Multiple choicecorenew
  The ground-state harmonic-oscillator wavefunction (in units where it equals $A e^{-x^2/2}$) has $A = \pi^{-1/4}$. What is the probability density $|\psi(0)|^2$ at $x=0$?
  beginner · physics · Probability Density |ψ|²
• Multiple choicecorenew
  An unnormalized wavefunction on $0 \le x \le 1$ is $\psi(x) = A x(1-x)$, and zero outside. Using $\int_0^1 x^2(1-x)^2 \, dx = 1/30$, find the normalization constant $A$ (take $A > 0$).
  beginner · physics · Normalization of ψ
• Multiple choicecorenew
  A qubit is in the state $\alpha|0\rangle + \beta|1\rangle$ with $|\alpha|^2 + |\beta|^2 = 1$. By the Born rule, the probability of measuring outcome 0 is:
  beginner · physics · The Born Rule
• Multiple choicecorenew
  An infinite-square-well ground state gives probability density $|\psi(x)|^2 = 2\sin^2(\pi x)$ on $0 \le x \le 1$. What is the probability of finding the particle in the interval $0 \le x \le 1/3$? Use $\int 2\sin^2(\pi x) \, dx = x - \sin(2\pi x)/(2\pi)$.
  beginner · physics · Probability in an Interval
• Multiple choicecorenew
  A particle on $[0,1]$ has probability density $|\psi(x)|^2 = 3x^2$. The expectation of position is $\langle x \rangle = \int_0^1 x(3x^2) \, dx$. What is $\langle x \rangle$?
  beginner · physics · Expectation of Position
• True / Falsecorenew
  Consider multiplying an entire wavefunction by a constant global phase factor $e^{i\theta}$ with $\theta$ real, giving $\psi' = e^{i\theta}\psi$. True or false: this global phase change leaves every measurable probability unchanged, so $\psi$ and $\psi'$ represent the same physical state.
  beginner · physics · Complex Phase of ψ
• Multiple choicecorenew
  Two orthonormal states psi1 and psi2 are combined as Psi = A(psi1 + psi2). Find the real positive normalization constant A that makes <Psi|Psi> = 1.
  beginner · physics · Superposition of Wavefunctions
• Multiple choicecorenew
  How do the state spaces of a single qubit and a particle moving on a line (the wavefunction case) differ?
  beginner · physics · Discrete vs Continuous States
• Multiple choicecorenew
  The probability density $\rho = |\psi|^2$ and the probability current $J$ obey the continuity equation derived from the Schrodinger equation. What is the correct form of this equation in one dimension?
  beginner · physics · Currents and Continuity (Preview)
• Multiple choicecorenew
  A wavefunction $\psi(x) = A e^{-x}$ for $x \ge 0$ is normalized so that $|\psi|^2 = 2 e^{-2x}$. The probability of finding the particle in $[0,1]$ is $P = \int_0^1 2 e^{-2x} \, dx = 1 - e^{-2}$. What is $P$?
  beginner · physics · Checkpoint: Working with ψ
• True / Falsecorenew
  A wave equation for quantum mechanics must preserve superposition: if $\psi_1$ and $\psi_2$ are valid states, then $c_1\psi_1 + c_2\psi_2$ must also be a valid state. True or false: this requirement forces the equation governing $\psi$ to be linear.
  beginner · physics · The Idea of a Wave Equation
• Multiple choicecorenew
  The time-dependent Schrodinger equation can be written compactly as i*hbar * d(Psi)/dt = H-hat * Psi. Because the equation is FIRST-order in time, what does this imply about the evolution of the wavefunction?
  beginner · physics · Time-Dependent Schrödinger Equation
• Multiple choicecorenew
  For a single particle of mass $m$ in one dimension under potential $V(x,t)$, the Hamiltonian operator $\hat{H}$ is built from the classical energy $E = p^2/(2m) + V$ by replacing momentum $p$ with the operator $\hat{p} = -i\hbar\,d/dx$. What explicit form results?
  beginner · physics · The Hamiltonian
• Multiple choicecorenew
  A stationary state has energy E = 1.0 eV (1 eV = 1.602e-19 J). Its phase evolves as e^{-i·E·t/hbar}. What is the angular frequency omega = E/hbar in rad/s? Use hbar = 1.055e-34 J·s.
  beginner · physics · Stationary States
• True / Falsecorenew
  Separating variables in the time-independent-potential Schrodinger equation gives $\Psi(x,t) = \psi(x)\,e^{-i E t/\hbar}$, a stationary state. True or false: for such a state the probability density $|\Psi(x,t)|^2$ is independent of time.
  beginner · physics · Time-Independent Schrödinger Equation
• Multiple choicecorenew
  In an infinite square well the ground-state energy is $E_1 = 0.38$ eV, and the levels scale as $E_n = n^2 \cdot E_1$. What is the energy of the $n=3$ level, $E_3$, in eV?
  beginner · physics · Energy Eigenvalues
• Multiple choicecorenew
  At a boundary $x = a$ where the potential $V$ is FINITE, which boundary conditions must the wavefunction $\psi$ satisfy when matching the solutions $\psi_L$ (left) and $\psi_R$ (right)?
  beginner · physics · Boundary Conditions
• Multiple choicecorenew
  A free electron has kinetic energy $K = 1$ eV $= 1.60218e-19$ J. Its momentum is $p = \sqrt{2 \cdot m \cdot K}$ and the matter wavenumber is $k = p/\hbar$. Using $m = 9.10938e-31$ kg and $\hbar = 1.05457e-34$ J$\cdot$s, what is $k$ in $m^{-1}$?
  beginner · physics · Free Particle Solutions
• True / Falsecorenew
  For a stationary state, the time-evolution phase factor is e^{-i·E·t/hbar}. Claim: the magnitude (absolute value) of this complex factor equals 1 for all times t.
  beginner · physics · Phase Evolution e^{-iEt/ħ}
• Multiple choicecorenew
  A stationary state has energy E = 2.0 eV (with 1 eV = 1.602e-19 J). Its phase rotates as e^{-i·E·t/hbar}. What is the angular frequency omega = E/hbar in rad/s? Use hbar = 1.055e-34 J·s.
  beginner · physics · Energy and Frequency
• Multiple choicecorenew
  For an electron in a 1 nm infinite square well, the ground-state energy is $E_1 = \pi^2\hbar^2/(2 m L^2) \approx 0.376$ eV. What is the energy of the $n=3$ stationary state, $E_3$?
  beginner · physics · Checkpoint: Stationary States
• Multiple choicecorenew
  In quantum mechanics, an observable physical quantity corresponds to a linear Hermitian operator. According to the operator postulate, what are the only possible outcomes of a measurement of that observable?
  beginner · physics · Observables as Operators
• Multiple choicecorenew
  For the state $|\psi\rangle = (|0\rangle + |1\rangle)/\sqrt{2}$, the Hermitian operator $Z$ (Pauli-Z) has eigenvalues $+1$ on $|0\rangle$ and $-1$ on $|1\rangle$. What is $\langle Z\rangle = |\alpha|^2 \cdot (+1) + |\beta|^2 \cdot (-1)$, with $|\alpha|^2 = |\beta|^2 = 1/2$?
  beginner · physics · Hermitian Operators
• Multiple choicecorenew
  A 2x2 Hermitian observable is $A = \begin{bmatrix}5 & 3 \\ 3 & 5\end{bmatrix}$. Its eigenvalues are the possible measurement outcomes. What is the larger eigenvalue?
  beginner · physics · Eigenvalues as Outcomes
• Multiple choicecorenew
  The operator $A = \begin{bmatrix}3 & 0 \\ 0 & -1\end{bmatrix}$ acts on the vector $v = |0\rangle = [1,0]$. Since $A \cdot v$ is a multiple of $v$, $|0\rangle$ is an eigenstate. What is its eigenvalue $\lambda$?
  beginner · physics · Eigenstates and Definite Values
• Multiple choicecorenew
  For the state $|\psi\rangle = (|0\rangle + |1\rangle)/\sqrt{2}$, what is the expectation value $\langle\sigma_z\rangle = \langle\psi|\sigma_z|\psi\rangle$, where $\sigma_z$ acts as $+1$ on $|0\rangle$ and $-1$ on $|1\rangle$?
  beginner · physics · Expectation Values ⟨A⟩
• Multiple choicecorenew
  In one-dimensional wave mechanics, what is the momentum operator $\hat{p}$ acting on a wave function $\psi(x)$ in the position representation?
  beginner · physics · The Position and Momentum Operators
• Multiple choicecorenew
  For the Pauli matrices $X = \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}$ and $Z = \begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}$, the commutator is $[X,Z] = XZ - ZX$. What is the value of the bottom-left (row 2, column 1) entry of $[X,Z]$, divided by 2?
  beginner · physics · Commutators
• Multiple choicecorenew
  Two Hermitian operators A-hat and B-hat are 'compatible' (simultaneously measurable, sharing a complete basis of simultaneous eigenstates) if and only if which condition holds?
  beginner · physics · Compatible Observables
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |0⟩?
  beginner · physics · Spin Observables (Preview)
• Numericcorenew
  For qubit 0 after this circuit, what is ⟨X⟩, the X-axis projection of its Bloch vector?
  beginner · physics · Spin Observables (Preview)
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 2521), how many shots land on |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  beginner · physics · Spin Observables (Preview)
• Multiple choicecorenew
  An observable $A$ has eigenvalues $+1$ and $-1$. On a given state, measurement gives $+1$ with probability $3/4$ and $-1$ with probability $1/4$. What is the standard deviation $\Delta A = \sqrt{\langle A^2\rangle - \langle A\rangle^2}$?
  beginner · physics · Measurement Statistics
• Multiple choicecorenew
  An operator $A = \begin{bmatrix}3 & 1 \\ 1 & 3\end{bmatrix}$ acts on the normalized state $|\psi\rangle = [1/\sqrt{5}, 2/\sqrt{5}]$. What is the expectation value $\langle\psi|A|\psi\rangle$?
  beginner · physics · Checkpoint: Operators and Outcomes
• Multiple choicecorenew
  The Heisenberg uncertainty principle for position and momentum states that the product of the standard deviations Delta x * Delta p can never fall below which value, for any state?
  beginner · physics · What Uncertainty Means
• Multiple choicecorenew
  An observable $A$ with eigenvalues $+1$ and $-1$ is measured: $+1$ occurs with probability $3/4$ and $-1$ with probability $1/4$. Find the standard deviation $\Delta A = \sqrt{\langle A^2\rangle - \langle A\rangle^2}$.
  beginner · physics · Standard Deviation of an Observable
• Multiple choicecorenew
  Using the Heisenberg relation Delta x * Delta p >= hbar/2 at the minimum-uncertainty bound, a wave packet has position uncertainty Delta x = 2.0 (in units where hbar = 1). What is the minimum momentum uncertainty Delta p?
  beginner · physics · Position–Momentum Uncertainty
• Multiple choicecorenew
  The Heisenberg uncertainty principle sets the lower bound Delta x * Delta p >= hbar/2. Using hbar = 1.0545718e-34 J·s, what is the numerical value of this bound hbar/2 (in J·s)?
  beginner · physics · The Commutator Bound
• True / Falsecorenew
  In ordinary (non-relativistic) quantum mechanics, time is a quantum observable represented by a Hermitian operator t-hat, exactly analogous to the position operator. True or false?
  beginner · physics · Energy–Time Uncertainty
• Multiple choicecorenew
  For a free-particle wave packet, if you confine it to a narrower window in position space (smaller spread Delta x), what necessarily happens to the spread of its momentum distribution Delta p?
  beginner · physics · Wave Packets and Spread
• Multiple choicecorenew
  Why are position x and momentum p called incompatible observables, with no quantum state being a simultaneous eigenstate of both?
  beginner · physics · Why You Cannot Know Both
• Multiple choicecorenew
  A spin-1/2 qubit is in state $|0\rangle$. For the Pauli observable $\sigma_x$, the expectation values are $\langle\sigma_x\rangle = 0$ and $\langle\sigma_x^2\rangle = 1$ (since $\sigma_x^2 = I$). What is the uncertainty $\Delta\sigma_x = \sqrt{\langle\sigma_x^2\rangle - \langle\sigma_x\rangle^2}$?
  beginner · physics · Uncertainty for Spin
• Multiple choicecorenew
  Which states are the minimum-uncertainty states that exactly saturate the Heisenberg bound sigma_x * sigma_p = hbar/2?
  beginner · physics · Minimum-Uncertainty States
• Multiple choicecorenew
  Observable A has eigenvalues +2 (prob 3/4) and -2 (prob 1/4); observable B has eigenvalues +1 (prob 1/4) and -1 (prob 3/4). Compute the product of the standard deviations, Delta A * Delta B.
  beginner · physics · Checkpoint: Uncertainty Calculations
• Multiple choicecorenew
  For a particle of mass $m$ in an infinite square well of width $L$, the allowed energy eigenvalues are $E_n = n^2 \pi^2 \hbar^2 / (2 m L^2)$ for $n = 1, 2, 3, \dots$. What does this imply about the lowest possible energy?
  beginner · physics · The Infinite Square Well
• Multiple choicecorenew
  For the infinite square well with walls at $x=0$ and $x=L$, the general solution inside is $\psi(x) = A\sin(kx) + B\cos(kx)$. After applying the boundary conditions $\psi(0)=0$ and $\psi(L)=0$, what results?
  beginner · physics · Boundary Conditions in the Well
• Multiple choicecorenew
  An electron is confined in a 1D infinite square well of width $L = 1$ nm, with $E_n = n^2 \pi^2 \hbar^2 / (2 m L^2)$. Using $\hbar=1.05457\times10^{-34}$ J·s, $m=9.10938\times10^{-31}$ kg, $1$ eV$=1.60218\times10^{-19}$ J, what is the $n = 2$ energy $E_2$ in eV?
  beginner · physics · Quantized Energy Levels
• Multiple choicecorenew
  The energy eigenfunctions of the 1D infinite square well are psi_n(x) = sin(n pi x / L). For the ground state n = 1 with L = 1, what is the value of psi_1(x) at the midpoint x = 0.5?
  beginner · physics · The Energy Eigenfunctions
• Multiple choicecorenew
  The energy eigenfunctions of the 1D infinite square well are $\psi_n(x) = N \sin(n \pi x / L)$. Normalization over $0$ to $L$ requires the integral of $|\psi_n|^2$ to equal 1. For $L = 1$, what is the normalization constant $N = \sqrt{2/L}$?
  beginner · physics · Normalizing the Eigenfunctions
• Multiple choicecorenew
  An electron is confined in a 1D infinite square well of width $L = 0.5$ nm. The ground-state (zero-point) energy is $E_1 = \pi^2 \hbar^2 / (2 m L^2)$. Using $\hbar=1.0546\times10^{-34}$ J·s, $m=9.1094\times10^{-31}$ kg, $1$ eV$=1.6022\times10^{-19}$ J, what is $E_1$ in eV?
  beginner · physics · Ground State and Zero-Point Energy
• Multiple choicecorenew
  For a particle in an infinite square well with eigenfunctions psi_n(x) ~ sin(n*pi*x/L), how many interior nodes (zeros strictly inside the well, excluding the walls) does the n-th state have?
  beginner · physics · Nodes and Quantum Numbers
• Multiple choicecorenew
  For an electron in the 1D infinite square well ground state ($n=1$, $L=1$), the probability density is $|\psi_1(x)|^2 = 2 \sin^2(\pi x)$. What is the probability of finding the particle in the left half of the well, i.e. the integral from $x=0$ to $x=0.5$?
  beginner · physics · Probability Distributions in the Well
• Multiple choicecorenew
  An electron in the infinite square well is in the superposition $\psi = c_1\psi_1 + c_2\psi_2$ with $c_1 = 3/5$. If a measurement is made, what is the probability of finding the particle in the $n=1$ energy eigenstate?
  beginner · physics · Superpositions in the Box
• Multiple choicecorenew
  An electron in a 1D infinite square well of width $L = 1$ nm is in an equal superposition of the $n=1$ and $n=2$ states, so $\langle E\rangle = (1/2)E_1 + (1/2)E_2$ with $E_n = n^2 \pi^2 \hbar^2/(2 m L^2)$. Using $\hbar=1.05457\times10^{-34}$ J·s, $m=9.10938\times10^{-31}$ kg, $1$ eV$=1.60218\times10^{-19}$ J, what is the expected energy $\langle E\rangle$ in eV?
  beginner · physics · Expectation Values in the Box
• Multiple choicecorenew
  An electron is confined in a 1D infinite square well of width $L = 1$ nm. The energy levels are $E_n = n^2 \pi^2 \hbar^2 / (2 m L^2)$. Using $\hbar = 1.0545718\times10^{-34}$ J·s, $m = 9.10938\times10^{-31}$ kg, and $1$ eV $= 1.60218\times10^{-19}$ J, what is the $n = 3$ energy $E_3$ in eV?
  beginner · physics · Checkpoint: Solve the Box
• True / Falsecorenew
  At a finite potential step, a particle with total energy E greater than the step height V_0 always transmits with certainty, exactly as classical physics predicts. True or false?
  beginner · physics · Finite Potential Step
• Multiple choicecorenew
  A particle with energy $E = 5$ eV hits a potential step of height $V_0 = 1$ eV ($E > V_0$, so it transmits over the step). With wavenumbers $k_1 \propto \sqrt{E}$ and $k_2 \propto \sqrt{E - V_0}$, the reflection coefficient is $R = ((k_1 - k_2)/(k_1 + k_2))^2$. What is $R$?
  beginner · physics · Reflection and Transmission
• True / Falsecorenew
  For a rectangular potential barrier of height V_0, a particle with energy E < V_0 (classically forbidden from the barrier) still has a nonzero probability of transmitting to the far side. True or false?
  beginner · physics · The Potential Barrier
• Multiple choicecorenew
  An electron ($m = 9.109\times10^{-31}$ kg) tunnels through a rectangular barrier of height $\Delta V = 1.0$ eV and width $a = 0.5$ nm. In the thick-barrier approximation the transmission is $T \approx e^{-2 \kappa a}$ with $\kappa = \sqrt{2 m \Delta V}/\hbar$ ($\hbar = 1.0546\times10^{-34}$ J·s, $1$ eV $= 1.602\times10^{-19}$ J). What is $T$?
  beginner · physics · Tunneling Through a Barrier
• Multiple choicecorenew
  An electron (mass $m_e = 9.109\times10^{-31}$ kg) faces a potential step of height $\Delta V = 1.0$ eV that exceeds its energy, so its wavefunction decays as $e^{-\kappa x}$ inside the barrier. The decay constant is $\kappa = \sqrt{2 m_e \Delta V}/\hbar$ ($\hbar = 1.0546\times10^{-34}$ J·s, $1$ eV $= 1.602\times10^{-19}$ J). What is $\kappa$, in m$^{-1}$?
  beginner · physics · Exponential Decay in the Barrier
• Multiple choicecorenew
  A particle of energy $E = 2.0$ eV hits a rectangular barrier of height $V_0 = 4.0$ eV and width $a = 3.0$ Å. With $\hbar^2/(2m) = 3.81$ eV·Å$^2$, the decay constant is $\kappa = \sqrt{(V_0-E)/(\hbar^2/2m)}$. Estimate the transmission using $T \approx \exp(-2 \kappa a)$.
  beginner · physics · Barrier Width and Probability
• Multiple choicecorenew
  In Gamow's model of alpha decay, an alpha particle with energy of a few MeV escapes a nucleus even though the Coulomb barrier peak is tens of MeV. By what mechanism does it escape, and how does the escape probability behave?
  beginner · physics · Alpha Decay as Tunneling
• Multiple choicecorenew
  In a scanning tunneling microscope, the tunneling current depends on the tip-sample gap width d through a factor e^(-2*kappa*d). For a typical metal work function (about 4.5 eV), roughly how much does the current change when the gap changes by 1 angstrom (10^-10 m)?
  beginner · physics · Scanning Tunneling Microscopy
• Multiple choicecorenew
  How does a finite (penetrable) square well of depth V_0 differ from an infinite square well of the same width regarding bound states?
  beginner · physics · The Finite Square Well
• Multiple choicecorenew
  A particle of energy $E = 1.0$ eV meets a barrier of height $V_0 = 3.0$ eV and width $a = 1.0$ Å. With $\hbar^2/(2m) = 3.81$ eV·Å$^2$, $\kappa = \sqrt{(V_0-E)/(\hbar^2/2m)}$. Using the exact transmission $T = 1 / (1 + V_0^2 \sinh^2(\kappa a) / (4 E (V_0 - E)))$, what is $T$?
  beginner · physics · Checkpoint: Tunneling Problems
• Multiple choicecorenew
  Why is the harmonic potential $V(x) = \tfrac{1}{2}m\omega^2 x^2$ considered a universal approximation for any smooth potential near a stable equilibrium point $x_0$?
  beginner · physics · The Harmonic Potential
• Multiple choicecorenew
  For a quantum harmonic oscillator with energy levels E_n = hbar·omega·(n + 1/2) and hbar·omega = 0.1 eV, what is the energy of the n = 3 level (in eV)?
  beginner · physics · Energy Levels Eₙ = ħω(n+½)
• Multiple choicecorenew
  The ground state (n = 0) of a quantum harmonic oscillator has nonzero energy E_0 = hbar·omega·(0 + 1/2). For hbar·omega = 0.1 eV, what is this zero-point energy (in eV)?
  beginner · physics · Zero-Point Energy
• Multiple choicecorenew
  The ground-state Gaussian of the quantum harmonic oscillator has position uncertainty $\Delta x = x_0/\sqrt{2}$ and momentum uncertainty $\Delta p = \hbar/(\sqrt{2}\,x_0)$. What is the uncertainty product $\Delta x \cdot \Delta p$, and what does it imply?
  beginner · physics · The Ground-State Gaussian
• Multiple choicecorenew
  Using the physicist's Hermite-polynomial recurrence H_{n+1}(u) = 2u*H_n(u) - 2n*H_{n-1}(u) starting from H_0(u) = 1 and H_1(u) = 2u, what is H_2(u)?
  beginner · physics · Hermite Polynomials (Overview)
• Multiple choicecorenew
  A quantum harmonic oscillator has equally spaced energy levels E_n = hbar·omega·(n + 1/2). If hbar·omega = 0.18 eV, what is the energy spacing between any two adjacent levels (E_{n+1} - E_n), in eV?
  beginner · physics · Equal Energy Spacing
• Multiple choicecorenew
  For a classical harmonic oscillator of amplitude A, where is the particle most likely to be found, i.e. where is the classical probability density P_cl(x) largest?
  beginner · physics · Classical vs Quantum Oscillator
• True / Falsecorenew
  In the quantized electromagnetic field, each field mode behaves as a harmonic oscillator, and adding one photon to a mode of frequency omega increases the field energy by exactly hbar*omega.
  beginner · physics · Photons as Oscillator Quanta
• Multiple choicecorenew
  For the quantum harmonic oscillator with energy levels E_n = hbar*omega*(n + 1/2), how does the fractional energy spacing Delta E / E_n between adjacent levels behave as the quantum number n grows large?
  beginner · physics · Correspondence at Large n
• Multiple choicecorenew
  A diatomic molecule's vibrations are modeled as a quantum harmonic oscillator with E_n = hbar·omega·(n + 1/2) and hbar·omega = 0.36 eV. What is the energy of the first excited state, n = 1 (in eV)?
  beginner · physics · Vibrations of Molecules
• Multiple choicecorenew
  A quantum harmonic oscillator has level energies E_n = hbar·omega·(n + 1/2) with hbar·omega = 0.25 eV. What is the energy difference E_5 - E_2 (in eV)?
  beginner · physics · Checkpoint: Oscillator Energies
• Multiple choicecorenew
  For an electron, which has spin quantum number s = 1/2, what is the magnitude of its spin angular momentum vector |S|?
  beginner · physics · Intrinsic Angular Momentum
• Multiple choicecorenew
  In the 1922 Stern-Gerlach experiment, a beam of silver atoms passed through an inhomogeneous magnetic field. Classical physics predicted a continuous smear, but what was actually observed, and what did it demonstrate?
  beginner · physics · The Stern–Gerlach Experiment
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|1⟩)?
  beginner · physics · Spin-½ States
• Multiple choicecorenew
  Using the Pauli operator $\sigma_x = \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}$ and the column vector $|{+x}\rangle = (1/\sqrt{2})\begin{bmatrix}1 \\ 1\end{bmatrix}$, compute the expectation value $\langle{+x}|\sigma_x|{+x}\rangle$. What is its value?
  beginner · physics · Pauli Matrices as Spin Operators
• Multiple choicecorenew
  A spin-1/2 particle is in state $|\psi\rangle = (3/5)|{\uparrow_z}\rangle + (4/5)|{\downarrow_z}\rangle$ (real amplitudes). When $S_z$ is measured, what is the probability of the $+\hbar/2$ (spin-up) outcome?
  beginner · physics · Measuring Spin Along z
• Multiple choicecorenew
  A spin-1/2 particle is prepared in the $S_z = +\hbar/2$ state $|{\uparrow_z}\rangle$. It is then measured along x, whose eigenstates are $|{+x}\rangle = (1/\sqrt{2})(|{\uparrow_z}\rangle + |{\downarrow_z}\rangle)$ and $|{-x}\rangle = (1/\sqrt{2})(|{\uparrow_z}\rangle - |{\downarrow_z}\rangle)$. What is the probability of obtaining $S_x = +\hbar/2$?
  beginner · physics · Measuring Spin Along x
• Multiple choicecorenew
  In a sequential Stern-Gerlach setup, atoms are filtered to S_z = +hbar/2, then passed through an x-analyzer keeping only S_x = +hbar/2, then through a z-analyzer. Since each cross-basis measurement gives 50/50, what fraction of the original spin-up atoms exit the final z-analyzer with S_z = +hbar/2?
  beginner · physics · Sequential Stern–Gerlach
• Multiple choicecorenew
  A spin-1/2 particle is in state $|\psi\rangle = \cos(\pi/6)|{\uparrow_z}\rangle + \sin(\pi/6)|{\downarrow_z}\rangle$, i.e. $\alpha = \sqrt{3}/2$ and $\beta = 1/2$. The expectation value of $S_z$ (in units of $\hbar/2$) is $\langle\sigma_z\rangle = |\alpha|^2 - |\beta|^2$. What is its value?
  beginner · physics · Spin Expectation Values
• Numericcorenew
  After running this circuit, what is the probability of measuring |0⟩?
  beginner · physics · Spin and the Bloch Sphere
• Numericcorenew
  Run this circuit. What is qubit 0's Bloch X-component — the expectation value ⟨X⟩?
  beginner · physics · Spin and the Bloch Sphere
• Predictcorenew
  On the seeded simulator (seed 8896), this circuit runs for 1024 shots. About how many come out |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  beginner · physics · Spin and the Bloch Sphere
• Multiple choicecorenew
  A spin-1/2 particle in a uniform field B = B_0*z-hat has Hamiltonian H = omega_0*S_z with energy eigenvalues E_± = ±hbar*omega_0/2. When a spin state is not aligned with the field, what does its expectation-value spin vector do?
  beginner · physics · Spin in a Magnetic Field
• Multiple choicecorenew
  An electron spin precesses in a magnetic field B = 1.0 T. Its Larmor angular frequency is omega_L = e·B/(2 m_e), with e = 1.602e-19 C and m_e = 9.109e-31 kg. What is omega_L, in rad/s?
  beginner · physics · Larmor Precession
• Multiple choicecorenew
  A spin-1/2 particle is in state $|\psi\rangle = (1/\sqrt{5})|{\uparrow_z}\rangle + (2/\sqrt{5})|{\downarrow_z}\rangle$. First $S_x$ is measured and the $+\hbar/2$ outcome is obtained (collapsing to $|{+x}\rangle = (1/\sqrt{2})(|{\uparrow_z}\rangle + |{\downarrow_z}\rangle)$), then $S_z$ is measured. With $\langle{+x}|\psi\rangle = 3/\sqrt{10}$ and $\langle{\uparrow_z}|{+x}\rangle = 1/\sqrt{2}$, what is the joint probability of getting $S_x = +\hbar/2$ followed by $S_z = +\hbar/2$?
  beginner · physics · Checkpoint: Spin Measurements
• True / Falsecorenew
  For a central potential $V(r)$ that depends only on the radial distance $r$, the angular-momentum operators $L^2$ and $L_z$ commute with the Hamiltonian $H = p^2/(2m) + V(r)$, making angular momentum a conserved quantity.
  beginner · physics · Central Potentials
• Multiple choicecorenew
  When the 3D Schrodinger equation for a central potential is separated as $\psi = R(r)\,Y(\theta,\phi)$, the azimuthal equation gives $\Phi(\phi) = \exp(i\,m_\ell\,\phi)$. What physical requirement forces $m_\ell$ to be an integer?
  beginner · physics · Separation in Spherical Coordinates
• Multiple choicecorenew
  The principal quantum number $n$ sets the hydrogen energy via $E_n = -13.6\ \text{eV} / n^2$. An electron occupying the $n = 3$ state has what energy (in eV)?
  beginner · physics · The Principal Quantum Number n
• Multiple choicecorenew
  For a fixed orbital angular-momentum quantum number ell, the magnetic quantum number m takes the integer values from -ell to +ell. How many distinct orbital orientations (values of m) exist for ell = 3?
  beginner · physics · Orbital Angular Momentum ℓ
• Multiple choicecorenew
  The magnetic quantum number m_ell labels the projection of orbital angular momentum and ranges over the integers -ell, ..., 0, ..., +ell. For a subshell with ell = 3 (an f subshell), how many allowed values of m_ell are there?
  beginner · physics · The Magnetic Quantum Number mₗ
• Multiple choicecorenew
  The bound-state energy levels of hydrogen are $E_n = -13.6\ \text{eV} / n^2$. What is the energy (in eV) of the $n = 3$ level?
  beginner · physics · Hydrogen Energy Levels
• Multiple choicecorenew
  In the hydrogen atom, the principal level n contains orbital angular-momentum subshells ell = 0, 1, ..., n-1, each contributing (2*ell + 1) distinct orbital states (ignoring electron spin). What is the total orbital degeneracy of the n = 3 level?
  beginner · physics · Degeneracy of Levels
• Multiple choicecorenew
  In the hydrogen atom, the shape of an orbital and its number of angular nodes are set by the azimuthal quantum number ell. How many angular nodes does a p orbital (ell = 1) have?
  beginner · physics · Atomic Orbitals s, p, d
• Multiple choicecorenew
  The Balmer series of hydrogen follows $1/\lambda = R_H (1/n_\text{lower}^2 - 1/n_\text{upper}^2)$ with $R_H = 1.09678 \times 10^7\ \text{m}^{-1}$ and $n_\text{lower} = 2$. For the H-alpha transition ($n_\text{upper} = 3$), what is the emitted wavelength in nanometers?
  beginner · physics · Spectral Series of Hydrogen
• Multiple choicecorenew
  Once electron spin (m_s = ±1/2) is added as the fourth quantum number, what is the degeneracy (number of distinct states) of the n-th energy level of hydrogen?
  beginner · physics · Spin and the Fourth Quantum Number
• True / Falsecorenew
  The Pauli exclusion principle states that no two electrons in the same atom can have an identical set of all four quantum numbers (n, ell, m_ell, m_s).
  beginner · physics · The Pauli Exclusion Principle (Preview)
• Multiple choicecorenew
  Including the two electron spin states, each orbital subshell ell of a hydrogen-like atom holds 2*(2*ell + 1) electrons. For the n = 3 shell (ell = 0, 1, 2), what is the maximum number of electrons it can hold?
  beginner · physics · Filling Atomic Shells
• Multiple choicecorenew
  Because the spin quantum number m_s has only two values, each spatial orbital (n, ell, m_ell) holds at most two electrons. Using this, how many electrons can the n = 3 shell hold at most?
  beginner · physics · Why Quantum Numbers Matter
• Multiple choicecorenew
  In hydrogen the energy levels are $E_n = -13.6\ \text{eV} / n^2$. An electron drops from $n = 4$ to $n = 2$, emitting a photon. Taking the photon energy as $E_\text{initial} - E_\text{final}$ ($E_4 - E_2$), what is its magnitude (in eV)?
  beginner · physics · Checkpoint: Hydrogen and Quantum Numbers
• Multiple choicecorenew
  According to this lesson, what is computation at its core?
  beginner · programming · What Is Computation?
• Multiple choicecorenew
  A single classical bit can store how many distinct values?
  beginner · programming · Bits: The Unit of Information
• Multiple choicecorenew
  Evaluate the boolean expression 0 OR (1 AND 1), where 0 = false and 1 = true. What is the result (as 0 or 1)?
  beginner · programming · Boolean Logic: AND, OR, NOT
• Multiple choicecorenew
  Treating XOR as a logic gate on bits, what is 1 XOR 0?
  beginner · programming · Logic Gates as Functions
• Multiple choicecorenew
  A truth table for a boolean function of 3 input bits has how many rows (one per input combination)?
  beginner · programming · Truth Tables
• True / Falsecorenew
  True or false: the classical AND gate is reversible (you can always recover both inputs from its single output bit).
  beginner · programming · Reversible vs Irreversible Gates
• Multiple choicecorenew
  Start with bit $b = 1$ and apply the NOT gate ($b \to 1 - b$) twice in a row. What is the final value of $b$?
  beginner · programming · The NOT Gate and Bit Flips
• Multiple choicecorenew
  A probabilistic bit is 1 with probability 0.7. What is the probability it is 0?
  beginner · programming · Probabilistic Bits
• True / Falsecorenew
  Per this lesson, a quantum computer is simply a faster classical computer that speeds up every problem.
  beginner · programming · Why Classical Computers Have Limits
• Multiple choicecorenew
  Per this lesson, what physical mechanism makes interference fundamentally different from ordinary probabilities in quantum computing?
  beginner · programming · Preview: What Makes Quantum Different
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |1⟩?
  beginner · programming · Introducing the Qubit
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|1⟩)?
  beginner · programming · The States |0⟩ and |1⟩
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |1⟩?
  beginner · programming · Your First Circuit
• Multiple choiceintronew
  A qubit is in the state $(|0\rangle + |1\rangle)/\sqrt{2}$. Which statement describes it best?
  beginner · programming · Superposition, Intuitively
• Numericcorenew
  After running this circuit, what is the expectation value $\langle X \rangle$ of qubit 0 (its Bloch X-component)?
  beginner · programming · Superposition, Intuitively
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 7563). How many of the shots come out $|0\rangle$? Estimate $P(|0\rangle) \times 1024$ and pick the closest option.
  beginner · programming · Superposition, Intuitively
• Multiple choicecorenew
  A qubit $\alpha|0\rangle + \beta|1\rangle$ has $\beta = 4/5$. What is the probability of measuring $|1\rangle$?
  beginner · programming · Amplitudes vs Probabilities
• True / Falsecorenew
  True or false: a qubit with amplitudes $\alpha = 1/\sqrt{2}$ and $\beta = 1/\sqrt{2}$ is properly normalized ($|\alpha|^2 + |\beta|^2 = 1$).
  beginner · programming · Normalization
• True / Falsecorenew
  Multiplying a qubit's whole state by a global phase $e^{i\theta}$ changes the probabilities you would measure.
  beginner · programming · Global Phase Is Unobservable
• Numericcorenew
  After running this circuit, what is the probability of measuring |0⟩?
  beginner · programming · Relative Phase Matters
• Numericcorenew
  Run this circuit. What is qubit 0's Bloch X-component — the expectation value ⟨X⟩?
  beginner · programming · Relative Phase Matters
• Predictcorenew
  On the seeded simulator (seed 9722), this circuit runs for 1024 shots. About how many come out |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  beginner · programming · Relative Phase Matters
• Multiple choicecorenew
  A simulator reports a qubit with amplitude $1/\sqrt{2}$ on the $|0\rangle$ basis state. What probability does that amplitude correspond to?
  beginner · programming · Reading the State Vector Output
• Multiple choicecorenew
  Per this lesson, how much extractable classical information does a single measurement of one qubit yield?
  beginner · programming · Qubit vs Bit: A Summary
• Multiple choicecorenew
  A qubit $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ has the amplitude of $|1\rangle$ equal to $1/2$. What is the probability of measuring $|1\rangle$?
  beginner · programming · Kets: |ψ⟩ Notation
• Multiple choicecorenew
  Compute the inner product $\langle 0|+\rangle$, where $|+\rangle = (|0\rangle + |1\rangle)/\sqrt{2}$. What is its value?
  beginner · programming · Bras and Inner Products
• Multiple choicecorenew
  In the orthonormal basis $\{|0\rangle, |1\rangle\}$, add the four inner products $\langle 0|0\rangle + \langle 0|1\rangle + \langle 1|0\rangle + \langle 1|1\rangle$. What is the sum?
  beginner · programming · Orthonormal Bases
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|0⟩)?
  beginner · programming · The |+⟩ and |−⟩ States
• Numericcorenew
  This circuit prepares qubit 0 in some state. What expectation value ⟨X⟩ does it have along the Bloch X-axis?
  beginner · programming · The |+⟩ and |−⟩ States
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 168). How many of the shots come out |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  beginner · programming · The |+⟩ and |−⟩ States
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |0⟩?
  beginner · programming · Changing Basis
• Numericcorenew
  For qubit 0 after this circuit, what is ⟨X⟩, the X-axis projection of its Bloch vector?
  beginner · programming · Changing Basis
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 3201), how many shots land on |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  beginner · programming · Changing Basis
• Multiple choicecorenew
  A complex amplitude is $z = 1/2 + i\sqrt{3}/2$. What is its magnitude $|z| = \sqrt{\text{re}^2 + \text{im}^2}$?
  beginner · programming · Amplitude as a Complex Number
• Multiple choicecorenew
  A qubit has amplitude $3/5$ on $|0\rangle$. What is the probability of measuring $|0\rangle$?
  beginner · programming · Probability from Amplitude
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|1⟩)?
  beginner · programming · Column Vectors in Code
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |0⟩?
  beginner · programming · Superposition with Arbitrary Weights
• Numericcorenew
  For qubit 0 after this circuit, what is ⟨X⟩, the X-axis projection of its Bloch vector?
  beginner · programming · Superposition with Arbitrary Weights
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 6697), how many shots land on |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  beginner · programming · Superposition with Arbitrary Weights
• Multiple choicecorenew
  In Dirac notation, the outer product $|\psi\rangle\langle\phi|$ produces what kind of object?
  beginner · programming · Dirac Notation Cheatsheet
• Numericcorenew
  After running this circuit, what is the probability of measuring |0⟩?
  beginner · programming · Checkpoint: Describe Any Single Qubit
• Numericcorenew
  Run this circuit. What is qubit 0's Bloch X-component — the expectation value ⟨X⟩?
  beginner · programming · Checkpoint: Describe Any Single Qubit
• Predictcorenew
  On the seeded simulator (seed 701), this circuit runs for 1024 shots. About how many come out |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  beginner · programming · Checkpoint: Describe Any Single Qubit
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|0⟩)?
  beginner · programming · Gates as Matrices
• Numericcorenew
  This circuit prepares qubit 0 in some state. What expectation value ⟨X⟩ does it have along the Bloch X-axis?
  beginner · programming · Gates as Matrices
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 2239). How many of the shots come out |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  beginner · programming · Gates as Matrices
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |1⟩?
  beginner · programming · The X (NOT) Gate
• Numericcorenew
  After running this circuit, what is the probability of measuring |0⟩?
  beginner · programming · The Z Gate
• Numericcorenew
  Run this circuit. What is qubit 0's Bloch X-component — the expectation value ⟨X⟩?
  beginner · programming · The Z Gate
• Predictcorenew
  On the seeded simulator (seed 8305), this circuit runs for 1024 shots. About how many come out |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  beginner · programming · The Z Gate
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |1⟩?
  beginner · programming · The Y Gate
• Multiple choicecorenew
  The Pauli matrices are $X = \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}$, $Y = \begin{bmatrix}0 & -i \\ i & 0\end{bmatrix}$, and $Z = \begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}$. Compute the SUM of their determinants: $\det(X) + \det(Y) + \det(Z)$.
  beginner · programming · Pauli Matrices
• Multiple choicecorenew
  A unitary gate preserves total probability. For the output state with amplitudes $a_0 = 1/\sqrt{2}$ and $a_1 = 1/\sqrt{2}$, what is $|a_0|^2 + |a_1|^2$?
  beginner · programming · Gates Are Unitary
• Numericcorenew
  After running this circuit, what is the probability of measuring |0⟩?
  beginner · programming · Inverse Gates
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |1⟩?
  beginner · programming · Composing Gates
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |0⟩?
  beginner · programming · X, Y, Z on Superpositions
• Numericcorenew
  After running this circuit, what is the expectation value ⟨X⟩ of qubit 0 (its Bloch X-component)?
  beginner · programming · X, Y, Z on Superpositions
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 2468). Predict how many give |0⟩. Estimate P(|0⟩) × 1024 and pick the closest option.
  beginner · programming · X, Y, Z on Superpositions
• Numericcorenew
  After running this circuit, what is the probability of measuring |1⟩?
  beginner · programming · Checkpoint: Build a Target State with Paulis
• Predictcorenew
  You apply the Hadamard gate to $|0\rangle$ and then measure in the computational basis. What is the probability of outcome 1?
  beginner · programming · The Hadamard Gate
• Buildcorenew
  Write a one-qubit circuit that turns $|0\rangle$ into an equal superposition — measuring it should give 0 and 1 each with probability 1/2.
  beginner · programming · The Hadamard Gate
• Numericcorenew
  After running this circuit, what is the expectation value $\langle X \rangle$ of qubit 0 (its Bloch X-component)?
  beginner · programming · The Hadamard Gate
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 5200). How many of the shots come out $|0\rangle$? Estimate $P(|0\rangle) \times 1024$ and pick the closest option.
  beginner · programming · The Hadamard Gate
• True / Falseintronew
  Applying the Hadamard gate twice in a row ($H \cdot H$) is equivalent to doing nothing — the identity.
  beginner · programming · Hadamard Twice Is Identity
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|0⟩)?
  beginner · programming · Creating |+⟩ and |−⟩ with H
• Numericcorenew
  This circuit prepares qubit 0 in some state. What expectation value ⟨X⟩ does it have along the Bloch X-axis?
  beginner · programming · Creating |+⟩ and |−⟩ with H
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 9107). How many of the shots come out |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  beginner · programming · Creating |+⟩ and |−⟩ with H
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |0⟩?
  beginner · programming · The S (Phase) Gate
• Numericcorenew
  For qubit 0 after this circuit, what is ⟨Y⟩, the Y-axis projection of its Bloch vector?
  beginner · programming · The S (Phase) Gate
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 2167), how many shots land on |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  beginner · programming · The S (Phase) Gate
• Numericcorenew
  After running this circuit, what is the probability of measuring |1⟩?
  beginner · programming · The T Gate
• Numericcorenew
  Run this circuit. What is qubit 0's Bloch X-component — the expectation value ⟨X⟩?
  beginner · programming · The T Gate
• Predictcorenew
  On the seeded simulator (seed 3041), this circuit runs for 1024 shots. About how many come out |1⟩? Estimate P(|1⟩) × 1024 and pick the closest option.
  beginner · programming · The T Gate
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |0⟩?
  beginner · programming · The General Phase Gate P(λ)
• Numericcorenew
  After running this circuit, what is the expectation value ⟨Y⟩ of qubit 0 (its Bloch Y-component)?
  beginner · programming · The General Phase Gate P(λ)
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 6074). Predict how many give |0⟩. Estimate P(|0⟩) × 1024 and pick the closest option.
  beginner · programming · The General Phase Gate P(λ)
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|0⟩)?
  beginner · programming · Phase Gates on |+⟩
• Numericcorenew
  This circuit prepares qubit 0 in some state. What expectation value ⟨Y⟩ does it have along the Bloch Y-axis?
  beginner · programming · Phase Gates on |+⟩
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 6948). How many of the shots come out |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  beginner · programming · Phase Gates on |+⟩
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |1⟩?
  beginner · programming · HZH = X and Friends
• Multiple choicecorenew
  Per this lesson, what phase does the S gate apply to the $|1\rangle$ component of a state?
  beginner · programming · S, T and Their Inverses
• Numericcorenew
  After running this circuit, what is the probability of measuring |0⟩?
  beginner · programming · Interference from Phases
• Numericcorenew
  Run this circuit. What is qubit 0's Bloch Y-component — the expectation value ⟨Y⟩?
  beginner · programming · Interference from Phases
• Predictcorenew
  On the seeded simulator (seed 6659), this circuit runs for 1024 shots. About how many come out |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  beginner · programming · Interference from Phases
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |1⟩?
  beginner · programming · Checkpoint: Phase Engineering
• Numericcorenew
  For qubit 0 after this circuit, what is ⟨X⟩, the X-axis projection of its Bloch vector?
  beginner · programming · Checkpoint: Phase Engineering
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 5785), how many shots land on |1⟩? Estimate P(|1⟩) × 1024 and pick the closest option.
  beginner · programming · Checkpoint: Phase Engineering
• Multiple choicecorenew
  On the Bloch sphere as described in this lesson, which computational basis state sits at the north pole ($\theta = 0$)?
  beginner · programming · Picturing a Qubit
• Numericcorenew
  After running this circuit, what is the probability of measuring |1⟩?
  beginner · programming · Poles: |0⟩ and |1⟩
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |0⟩?
  beginner · programming · The Equator: |+⟩, |−⟩, |i⟩, |−i⟩
• Numericcorenew
  After running this circuit, what is the expectation value ⟨Y⟩ of qubit 0 (its Bloch Y-component)?
  beginner · programming · The Equator: |+⟩, |−⟩, |i⟩, |−i⟩
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 5663). Predict how many give |0⟩. Estimate P(|0⟩) × 1024 and pick the closest option.
  beginner · programming · The Equator: |+⟩, |−⟩, |i⟩, |−i⟩
• Numericcorenew
  Prepare $|1\rangle$ by applying X to $|0\rangle$. What is the Bloch $z$-coordinate $\langle Z \rangle$ of the resulting qubit?
  beginner · programming · Bloch Coordinates (x, y, z)
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |1⟩?
  beginner · programming · Gates as Rotations
• Numericcorenew
  After running this circuit, what is the probability of measuring |0⟩?
  beginner · programming · Hadamard as a Rotation
• Numericcorenew
  Run this circuit. What is qubit 0's Bloch X-component — the expectation value ⟨X⟩?
  beginner · programming · Hadamard as a Rotation
• Predictcorenew
  On the seeded simulator (seed 6126), this circuit runs for 1024 shots. About how many come out |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  beginner · programming · Hadamard as a Rotation
• Multiple choicecorenew
  Per this lesson, viewing Z, S, and T as rotations about the z-axis, which composition identity holds?
  beginner · programming · Phase Gates as Z-Rotations
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |1⟩?
  beginner · programming · Reading the Bloch Display
• Numericcorenew
  After running this circuit, what is the expectation value ⟨X⟩ of qubit 0 (its Bloch X-component)?
  beginner · programming · Reading the Bloch Display
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 4535). Predict how many give |1⟩. Estimate P(|1⟩) × 1024 and pick the closest option.
  beginner · programming · Reading the Bloch Display
• Multiple choicecorenew
  Per this lesson, where on the Bloch sphere do pure states live, and where do mixed states live?
  beginner · programming · Pure vs Mixed (Preview)
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |1⟩?
  beginner · programming · Checkpoint: Navigate the Bloch Sphere
• Numericcorenew
  For qubit 0 after this circuit, what is ⟨X⟩, the X-axis projection of its Bloch vector?
  beginner · programming · Checkpoint: Navigate the Bloch Sphere
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 628), how many shots land on |1⟩? Estimate P(|1⟩) × 1024 and pick the closest option.
  beginner · programming · Checkpoint: Navigate the Bloch Sphere
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |0⟩?
  beginner · programming · The Measurement Postulate
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|0⟩)?
  beginner · programming · Born Rule in Practice
• Numericcorenew
  This circuit prepares qubit 0 in some state. What expectation value ⟨X⟩ does it have along the Bloch X-axis?
  beginner · programming · Born Rule in Practice
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 5375). How many of the shots come out |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  beginner · programming · Born Rule in Practice
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |0⟩?
  beginner · programming · Measuring |+⟩
• Numericcorenew
  For qubit 0 after this circuit, what is ⟨X⟩, the X-axis projection of its Bloch vector?
  beginner · programming · Measuring |+⟩
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 8408), how many shots land on |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  beginner · programming · Measuring |+⟩
• Numericcorenew
  After running this circuit, what is the probability of measuring |0⟩?
  beginner · programming · Shots and Sampling
• Numericcorenew
  Run this circuit. What is qubit 0's Bloch X-component — the expectation value ⟨X⟩?
  beginner · programming · Shots and Sampling
• Predictcorenew
  On the seeded simulator (seed 9282), this circuit runs for 1024 shots. About how many come out |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  beginner · programming · Shots and Sampling
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |00⟩?
  beginner · programming · Histograms of Outcomes
• Numericcorenew
  After running this circuit, what is the expectation value ⟨X⟩ of qubit 0 (its Bloch X-component)?
  beginner · programming · Histograms of Outcomes
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 2342). Predict how many give |00⟩. Estimate P(|00⟩) × 1024 and pick the closest option.
  beginner · programming · Histograms of Outcomes
• Numericcorenew
  For this circuit, what is the probability that measuring all 2 qubits yields exactly |01⟩ (qubit 0 leftmost)?
  beginner · programming · Histograms of Outcomes
• Multiple choicecorenew
  A qubit in state $\alpha|0\rangle + \beta|1\rangle$ is measured in the computational basis and yields outcome 0. Per this lesson, what is the qubit's state immediately afterward?
  beginner · programming · Collapse After Measurement
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |1⟩?
  beginner · programming · Measuring in the X Basis
• Multiple choicecorenew
  A qubit is prepared in the state $|+\rangle = (|0\rangle + |1\rangle)/\sqrt{2}$. In the ideal (infinite-shot) limit, what is the probability of measuring outcome $|0\rangle$ in the computational basis?
  beginner · programming · Statistics vs Single Shots
• Multiple choicecorenew
  When estimating a probability $p$ from $N$ measurement shots, the standard error is at most $SE = 1/(2\sqrt{N})$. What is the minimum number of shots $N$ needed so that $SE \le 0.01$?
  beginner · programming · Estimating Probabilities Accurately
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |00⟩?
  beginner · programming · Checkpoint: Predict Then Measure
• Predictcorenew
  On the seeded simulator (seed 5034), this circuit runs for 1024 shots. About how many come out |00⟩? Estimate P(|00⟩) × 1024 and pick the closest option.
  beginner · programming · Checkpoint: Predict Then Measure
• Numericcorenew
  Run this circuit, then measure all 2 qubits at once. What is the probability of the joint outcome |11⟩ (qubit 0 leftmost)?
  beginner · programming · Checkpoint: Predict Then Measure
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |0⟩?
  beginner · programming · Rotations RX, RY, RZ
• Numericcorenew
  For qubit 0 after this circuit, what is ⟨X⟩, the X-axis projection of its Bloch vector?
  beginner · programming · Rotations RX, RY, RZ
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 9567), how many shots land on |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  beginner · programming · Rotations RX, RY, RZ
• Numericcorenew
  After running this circuit, what is the probability of measuring |1⟩?
  beginner · programming · RX and the X Axis
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |0⟩?
  beginner · programming · RY for Real Amplitudes
• Numericcorenew
  After running this circuit, what is the expectation value ⟨X⟩ of qubit 0 (its Bloch X-component)?
  beginner · programming · RY for Real Amplitudes
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 5660). Predict how many give |0⟩. Estimate P(|0⟩) × 1024 and pick the closest option.
  beginner · programming · RY for Real Amplitudes
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|0⟩)?
  beginner · programming · RZ and Phase
• Numericcorenew
  This circuit prepares qubit 0 in some state. What expectation value ⟨X⟩ does it have along the Bloch X-axis?
  beginner · programming · RZ and Phase
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 8693). How many of the shots come out |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  beginner · programming · RZ and Phase
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |1⟩?
  beginner · programming · Angle Conventions (θ/2)
• Numericcorenew
  After running this circuit, what is the probability of measuring |0⟩?
  beginner · programming · Building H from Rotations
• Numericcorenew
  Run this circuit. What is qubit 0's Bloch X-component — the expectation value ⟨X⟩?
  beginner · programming · Building H from Rotations
• Predictcorenew
  On the seeded simulator (seed 4786), this circuit runs for 1024 shots. About how many come out |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  beginner · programming · Building H from Rotations
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |0⟩?
  beginner · programming · Any Single-Qubit Unitary
• Numericcorenew
  After running this circuit, what is the expectation value ⟨X⟩ of qubit 0 (its Bloch X-component)?
  beginner · programming · Any Single-Qubit Unitary
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 7819). Predict how many give |0⟩. Estimate P(|0⟩) × 1024 and pick the closest option.
  beginner · programming · Any Single-Qubit Unitary
• Multiple choicecorenew
  Per this lesson, what does it mean for a gate set to be universal for single-qubit operations?
  beginner · programming · Universal Gate Sets (Single Qubit)
• Numericcorenew
  After running this circuit, what is the probability of measuring |0⟩?
  beginner · programming · Preparing an Arbitrary State
• Numericcorenew
  Run this circuit. What is qubit 0's Bloch Y-component — the expectation value ⟨Y⟩?
  beginner · programming · Preparing an Arbitrary State
• Predictcorenew
  On the seeded simulator (seed 5777), this circuit runs for 1024 shots. About how many come out |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  beginner · programming · Preparing an Arbitrary State
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |0⟩?
  beginner · programming · Checkpoint: State Preparation
• Numericcorenew
  For qubit 0 after this circuit, what is ⟨Y⟩, the Y-axis projection of its Bloch vector?
  beginner · programming · Checkpoint: State Preparation
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 4903), how many shots land on |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  beginner · programming · Checkpoint: State Preparation
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |11⟩?
  beginner · programming · Two-Qubit States
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|01⟩)?
  beginner · programming · The Tensor Product
• Numericcorenew
  This circuit prepares qubit 0 in some state. What expectation value ⟨X⟩ does it have along the Bloch X-axis?
  beginner · programming · The Tensor Product
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 5585). How many of the shots come out |01⟩? Estimate P(|01⟩) × 1024 and pick the closest option.
  beginner · programming · The Tensor Product
• Numericcorenew
  Run this circuit, then measure all 2 qubits at once. What is the probability of the joint outcome |11⟩ (qubit 0 leftmost)?
  beginner · programming · The Tensor Product
• Multiple choicecorenew
  Using the convention where qubit 0 is the MOST significant bit, the two-qubit basis state |10> has qubit 0 = 1 and qubit 1 = 0. What is its integer index in the state vector (value = q0*2^1 + q1*2^0)?
  beginner · programming · Basis Ordering and Indexing
• Numericcorenew
  After running this circuit, what is the probability of measuring |01⟩?
  beginner · programming · Product States
• Numericcorenew
  Run this circuit. What is qubit 0's Bloch X-component — the expectation value ⟨X⟩?
  beginner · programming · Product States
• Predictcorenew
  On the seeded simulator (seed 9492), this circuit runs for 1024 shots. About how many come out |01⟩? Estimate P(|01⟩) × 1024 and pick the closest option.
  beginner · programming · Product States
• Numericcorenew
  After this circuit, what is P(|11⟩) for the whole 2-qubit register measured together (qubit 0 leftmost)?
  beginner · programming · Product States
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |01⟩?
  beginner · programming · Independent Gates on Each Qubit
• Numericcorenew
  After running this circuit, what is the expectation value ⟨X⟩ of qubit 0 (its Bloch X-component)?
  beginner · programming · Independent Gates on Each Qubit
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 2552). Predict how many give |01⟩. Estimate P(|01⟩) × 1024 and pick the closest option.
  beginner · programming · Independent Gates on Each Qubit
• Numericcorenew
  For this circuit, what is the probability that measuring all 2 qubits yields exactly |11⟩ (qubit 0 leftmost)?
  beginner · programming · Independent Gates on Each Qubit
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|00⟩)?
  beginner · programming · Superposition Across Two Qubits
• Numericcorenew
  This circuit prepares qubit 0 in some state. What expectation value ⟨X⟩ does it have along the Bloch X-axis?
  beginner · programming · Superposition Across Two Qubits
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 2089). How many of the shots come out |00⟩? Estimate P(|00⟩) × 1024 and pick the closest option.
  beginner · programming · Superposition Across Two Qubits
• Numericcorenew
  Run this circuit, then measure all 2 qubits at once. What is the probability of the joint outcome |01⟩ (qubit 0 leftmost)?
  beginner · programming · Superposition Across Two Qubits
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |00⟩?
  beginner · programming · Measuring One of Two Qubits
• Multiple choicecorenew
  A two-qubit state is written as $c_{00}|00\rangle + c_{01}|01\rangle + c_{10}|10\rangle + c_{11}|11\rangle$ with $c_{00} = 1/\sqrt{2}$, $c_{11} = 1/\sqrt{2}$, and $c_{01} = c_{10} = 0$. What is the marginal probability that qubit 0 is measured as 0, $P(q0=0) = |c_{00}|^2 + |c_{01}|^2$?
  beginner · programming · Marginal Probabilities
• Multiple choicecorenew
  Per this lesson, how many complex amplitudes does the full state vector of an n-qubit register hold?
  beginner · programming · Counting Amplitudes Grows Fast
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |00⟩?
  beginner · programming · Checkpoint: Two-Qubit Product States
• Numericcorenew
  For qubit 0 after this circuit, what is ⟨X⟩, the X-axis projection of its Bloch vector?
  beginner · programming · Checkpoint: Two-Qubit Product States
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 3750), how many shots land on |00⟩? Estimate P(|00⟩) × 1024 and pick the closest option.
  beginner · programming · Checkpoint: Two-Qubit Product States
• Numericcorenew
  After running this circuit, what is the probability that all 2 qubits together read |01⟩ (qubit 0 leftmost)?
  beginner · programming · Checkpoint: Two-Qubit Product States
• Multiple choicecorenew
  Alice and Bob share an entangled pair. Alice measures her qubit in New York; Bob holds the other in Tokyo. Which statement is correct?
  beginner · programming · What Is Entanglement?
• Multiple choicecorenew
  You measure BOTH qubits of the Bell state $(|00\rangle + |11\rangle)/\sqrt{2}$ in the computational basis. Which joint outcomes can occur? (Select all that apply.)
  beginner · programming · The First Bell State
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 1399). How many of the shots come out $|00\rangle$? Estimate $P(|00\rangle) \times 1024$ and pick the closest option.
  beginner · programming · The First Bell State
• Numericcorenew
  After running this circuit, what is the probability that all 2 qubits together read $|11\rangle$ (qubit 0 leftmost)?
  beginner · programming · The First Bell State
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|01⟩)?
  beginner · programming · All Four Bell States
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 8339). Predict how many give |01⟩. Estimate P(|01⟩) × 1024 and pick the closest option.
  beginner · programming · All Four Bell States
• Numericcorenew
  After running this circuit, what is the probability that all 2 qubits together read |10⟩ (qubit 0 leftmost)?
  beginner · programming · All Four Bell States
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |00⟩?
  beginner · programming · Correlated Measurements
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 5306). How many of the shots come out |00⟩? Estimate P(|00⟩) × 1024 and pick the closest option.
  beginner · programming · Correlated Measurements
• Numericcorenew
  After this circuit, what is P(|11⟩) for the whole 2-qubit register measured together (qubit 0 leftmost)?
  beginner · programming · Correlated Measurements
• True / Falsecorenew
  Per this lesson, two parties sharing a Bell pair can use their entanglement alone to send a message faster than light, with no classical channel needed.
  beginner · programming · Entangled ≠ Communicating
• True / Falsecorenew
  A two-qubit state $\alpha|00\rangle + \beta|01\rangle + \gamma|10\rangle + \delta|11\rangle$ is a product (separable) state only if $\alpha\delta = \beta\gamma$. For the Bell state with $\alpha = 1/\sqrt{2}$, $\delta = 1/\sqrt{2}$, and $\beta = \gamma = 0$, the state is ENTANGLED. True or false?
  beginner · programming · Product vs Entangled: A Test
• Multiple choicecorenew
  Per this lesson, the Bell state |Phi+> is a pure two-qubit state. What is the reduced state of just one of its qubits (obtained by the partial trace)?
  beginner · programming · Reduced States Are Mixed
• Multiple choicecorenew
  When you display either qubit of the Bell state $|\Phi+\rangle = (|00\rangle + |11\rangle)/\sqrt{2}$ on its own Bloch sphere, where does its arrow point, per this lesson?
  beginner · programming · Bell States in the Bloch Picture
• Numericcorenew
  After running this circuit, what is the probability of measuring |000⟩?
  beginner · programming · Creating Entanglement on Demand
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 748), how many shots land on |000⟩? Estimate P(|000⟩) × 1024 and pick the closest option.
  beginner · programming · Creating Entanglement on Demand
• Numericcorenew
  For this circuit, what is the probability that measuring all 3 qubits yields exactly |011⟩ (qubit 0 leftmost)?
  beginner · programming · Creating Entanglement on Demand
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |000⟩?
  beginner · programming · GHZ: Three-Qubit Entanglement
• Predictcorenew
  On the seeded simulator (seed 9847), this circuit runs for 1024 shots. About how many come out |000⟩? Estimate P(|000⟩) × 1024 and pick the closest option.
  beginner · programming · GHZ: Three-Qubit Entanglement
• Numericcorenew
  Run this circuit, then measure all 3 qubits at once. What is the probability of the joint outcome |111⟩ (qubit 0 leftmost)?
  beginner · programming · GHZ: Three-Qubit Entanglement
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|00⟩)?
  beginner · programming · Checkpoint: Engineer an Entangled State
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 6814). Predict how many give |00⟩. Estimate P(|00⟩) × 1024 and pick the closest option.
  beginner · programming · Checkpoint: Engineer an Entangled State
• Numericcorenew
  After running this circuit, what is the probability that all 2 qubits together read |11⟩ (qubit 0 leftmost)?
  beginner · programming · Checkpoint: Engineer an Entangled State
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|11⟩)?
  beginner · programming · The CNOT Gate
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |11⟩?
  beginner · programming · Control and Target
• Numericcorenew
  After running this circuit, what is the probability of measuring |00⟩?
  beginner · programming · CNOT on Superpositions
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 9220), how many shots land on |00⟩? Estimate P(|00⟩) × 1024 and pick the closest option.
  beginner · programming · CNOT on Superpositions
• Numericcorenew
  For this circuit, what is the probability that measuring all 2 qubits yields exactly |11⟩ (qubit 0 leftmost)?
  beginner · programming · CNOT on Superpositions
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |01⟩?
  beginner · programming · The Controlled-Z Gate
• Numericcorenew
  After running this circuit, what is the expectation value ⟨X⟩ of qubit 0 (its Bloch X-component)?
  beginner · programming · The Controlled-Z Gate
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 2280). Predict how many give |01⟩. Estimate P(|01⟩) × 1024 and pick the closest option.
  beginner · programming · The Controlled-Z Gate
• Numericcorenew
  For this circuit, what is the probability that measuring all 2 qubits yields exactly |11⟩ (qubit 0 leftmost)?
  beginner · programming · The Controlled-Z Gate
• Multiple choicecorenew
  Per this lesson, how is a CNOT (control qubit 0, target qubit 1) built from a CZ gate and Hadamards?
  beginner · programming · CNOT from CZ and Hadamards
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |01⟩?
  beginner · programming · The SWAP Gate
• Multiple choicecorenew
  The SWAP gate can be built from exactly three CNOTs with alternating control and target. Which sequence of CNOTs (each written control->target) realizes SWAP on qubits 0 and 1?
  beginner · programming · SWAP from Three CNOTs
• Multiple choicecorenew
  A controlled-phase gate $CP(\lambda)$ acts on two qubits. Which single basis state does it multiply by the phase $e^{i\lambda}$, leaving all other basis states unchanged?
  beginner · programming · Controlled Phase Gates
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |01⟩?
  beginner · programming · Phase Kickback
• Numericcorenew
  For qubit 0 after this circuit, what is ⟨X⟩, the X-axis projection of its Bloch vector?
  beginner · programming · Phase Kickback
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 4972), how many shots land on |01⟩? Estimate P(|01⟩) × 1024 and pick the closest option.
  beginner · programming · Phase Kickback
• Numericcorenew
  After running this circuit, what is the probability that all 2 qubits together read |11⟩ (qubit 0 leftmost)?
  beginner · programming · Phase Kickback
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|00⟩)?
  beginner · programming · Checkpoint: Controlled-Gate Puzzles
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 1939). Predict how many give |00⟩. Estimate P(|00⟩) × 1024 and pick the closest option.
  beginner · programming · Checkpoint: Controlled-Gate Puzzles
• Numericcorenew
  After running this circuit, what is the probability that all 2 qubits together read |11⟩ (qubit 0 leftmost)?
  beginner · programming · Checkpoint: Controlled-Gate Puzzles
• Multiple choicecorenew
  In a quantum circuit diagram, in which direction does time flow, so that gates are applied in that order?
  beginner · programming · Circuit Diagrams
• Multiple choicecorenew
  A circuit on a single qubit applies the gates H, then Z, then H in sequence (each gate acts on the same qubit, one after another). Since no gates can run in parallel on one qubit, what is the circuit depth?
  beginner · programming · Circuit Depth and Width
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|0⟩)?
  beginner · programming · Order of Operations
• Numericcorenew
  This circuit prepares qubit 0 in some state. What expectation value ⟨X⟩ does it have along the Bloch X-axis?
  beginner · programming · Order of Operations
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 593). How many of the shots come out |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  beginner · programming · Order of Operations
• True / Falsecorenew
  True or false: A barrier corresponds to a physical quantum operation that changes the state vector of the circuit.
  beginner · programming · Barriers and Structure
• Numericcorenew
  After running this circuit, what is the probability of measuring |00⟩?
  beginner · programming · Translating Math to Circuits
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 2341), how many shots land on |00⟩? Estimate P(|00⟩) × 1024 and pick the closest option.
  beginner · programming · Translating Math to Circuits
• Numericcorenew
  For this circuit, what is the probability that measuring all 2 qubits yields exactly |11⟩ (qubit 0 leftmost)?
  beginner · programming · Translating Math to Circuits
• Multiple choicecorenew
  A circuit applies gate G1 first, then G2, then G3. What is the single unitary matrix U for the whole circuit?
  beginner · programming · Translating Circuits to Math
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |0000⟩?
  beginner · programming · Reusing Subcircuits
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 5839). How many of the shots come out |0000⟩? Estimate P(|0000⟩) × 1024 and pick the closest option.
  beginner · programming · Reusing Subcircuits
• Numericcorenew
  After this circuit, what is P(|0011⟩) for the whole 4-qubit register measured together (qubit 0 leftmost)?
  beginner · programming · Reusing Subcircuits
• Multiple choicecorenew
  The T gate advances the phase of $|1\rangle$ by $\pi/4$. Applying T twice ($T^2$) is circuit-equivalent to which single gate?
  beginner · programming · Equivalent Circuits
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |00⟩?
  beginner · programming · Debugging a Circuit
• Predictcorenew
  On the seeded simulator (seed 1932), this circuit runs for 1024 shots. About how many come out |00⟩? Estimate P(|00⟩) × 1024 and pick the closest option.
  beginner · programming · Debugging a Circuit
• Numericcorenew
  Run this circuit, then measure all 2 qubits at once. What is the probability of the joint outcome |11⟩ (qubit 0 leftmost)?
  beginner · programming · Debugging a Circuit
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|01⟩)?
  beginner · programming · Checkpoint: From Spec to Circuit
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 8872). Predict how many give |01⟩. Estimate P(|01⟩) × 1024 and pick the closest option.
  beginner · programming · Checkpoint: From Spec to Circuit
• Numericcorenew
  After running this circuit, what is the probability that all 2 qubits together read |10⟩ (qubit 0 leftmost)?
  beginner · programming · Checkpoint: From Spec to Circuit
• True / Falsecorenew
  There exists a single quantum operation that perfectly copies an arbitrary, unknown qubit state $|\psi\rangle$ onto a second qubit.
  beginner · programming · The No-Cloning Theorem
• Multiple choicecorenew
  Quantum teleportation transfers an unknown qubit state from Alice to Bob. Which two resources does the protocol consume to do this?
  beginner · programming · Teleportation: The Idea
• Numericcorenew
  After running this circuit, what is the probability of measuring |00⟩?
  beginner · programming · Sharing a Bell Pair
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 8719), how many shots land on |00⟩? Estimate P(|00⟩) × 1024 and pick the closest option.
  beginner · programming · Sharing a Bell Pair
• Numericcorenew
  For this circuit, what is the probability that measuring all 2 qubits yields exactly |11⟩ (qubit 0 leftmost)?
  beginner · programming · Sharing a Bell Pair
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |00⟩?
  beginner · programming · The Bell Measurement
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|001⟩)?
  beginner · programming · Classical Correction
• Numericcorenew
  This circuit prepares qubit 0 in some state. What expectation value ⟨X⟩ does it have along the Bloch X-axis?
  beginner · programming · Classical Correction
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 8308). How many of the shots come out |001⟩? Estimate P(|001⟩) × 1024 and pick the closest option.
  beginner · programming · Classical Correction
• Numericcorenew
  Run this circuit, then measure all 3 qubits at once. What is the probability of the joint outcome |011⟩ (qubit 0 leftmost)?
  beginner · programming · Classical Correction
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |000⟩?
  beginner · programming · Full Teleportation Circuit
• Numericcorenew
  For qubit 0 after this circuit, what is ⟨X⟩, the X-axis projection of its Bloch vector?
  beginner · programming · Full Teleportation Circuit
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 1368), how many shots land on |000⟩? Estimate P(|000⟩) × 1024 and pick the closest option.
  beginner · programming · Full Teleportation Circuit
• Numericcorenew
  After running this circuit, what is the probability that all 3 qubits together read |001⟩ (qubit 0 leftmost)?
  beginner · programming · Full Teleportation Circuit
• Multiple choicecorenew
  In quantum teleportation, before Bob receives Alice's 2 classical bits, what is the state of his qubit (its reduced density matrix)?
  beginner · programming · Why Teleportation Obeys No-Signalling
• Multiple choicecorenew
  In superdense coding, Alice and Bob share one Bell pair in advance. By sending Bob just one qubit, how many classical bits can Alice communicate?
  beginner · programming · Superdense Coding: The Idea
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |00⟩?
  beginner · programming · Encoding Two Bits
• Predictcorenew
  On the seeded simulator (seed 4471), this circuit runs for 1024 shots. About how many come out |00⟩? Estimate P(|00⟩) × 1024 and pick the closest option.
  beginner · programming · Encoding Two Bits
• Numericcorenew
  Run this circuit, then measure all 2 qubits at once. What is the probability of the joint outcome |11⟩ (qubit 0 leftmost)?
  beginner · programming · Encoding Two Bits
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|10⟩)?
  beginner · programming · Decoding with a Bell Measurement
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |11⟩?
  beginner · programming · Checkpoint: Teleport and Code
• Multiple choicecorenew
  The standard reversible oracle embedding writes f into an ancilla register via U_f|x>|y> = |x>|y XOR f(x)>. What happens when you apply U_f twice in a row?
  beginner · programming · Oracles and Black Boxes
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|00⟩)?
  beginner · programming · Quantum Parallelism
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 6288). Predict how many give |00⟩. Estimate P(|00⟩) × 1024 and pick the closest option.
  beginner · programming · Quantum Parallelism
• Numericcorenew
  After running this circuit, what is the probability that all 2 qubits together read |11⟩ (qubit 0 leftmost)?
  beginner · programming · Quantum Parallelism
• Multiple choicecorenew
  A phase oracle implements $|x\rangle \to (-1)^{f(x)} |x\rangle$. For the single-bit identity function $f(x)=x$ (so $f(0)=0$, $f(1)=1$), which single-qubit gate is the phase oracle?
  beginner · programming · The Phase Oracle
• True / Falsecorenew
  True or false: A balanced Boolean function f on n bits outputs 0 on exactly 2^(n-1) of the inputs and 1 on the remaining 2^(n-1).
  beginner · programming · Constant vs Balanced Functions
• True / Falsecorenew
  For the Deutsch problem (deciding whether a one-bit function $f:\{0,1\}\to\{0,1\}$ is constant or balanced), a classical algorithm needs two oracle queries in the worst case because a single query never determines the class.
  beginner · programming · The Deutsch Problem
• Multiple choicecorenew
  In Deutsch's algorithm, after the final Hadamard you measure the input qubit (qubit 0). Which outcome certifies that f is balanced (f(0) != f(1))?
  beginner · programming · Deutsch’s Algorithm
• Multiple choicecorenew
  The final Hadamard in Deutsch's algorithm maps the register qubit to a computational basis state. What are $H|+\rangle$ and $H|-\rangle$?
  beginner · programming · Interference Does the Work
• Multiple choicecorenew
  For an n-bit function promised to be constant or balanced, how many oracle queries does the Deutsch-Jozsa algorithm need to decide which, compared with the classical worst case?
  beginner · programming · Scaling Up: Deutsch–Jozsa
• Multiple choicecorenew
  To build the oracle U_f for the balanced function f(x0,x1) = x0 XOR x1 on qubits 0,1 (inputs) and qubit 2 (ancilla), which gates write the function value into the ancilla?
  beginner · programming · Building the DJ Oracle
• Multiple choicecorenew
  In the Deutsch-Jozsa algorithm, after the final layer of Hadamards you measure the n input qubits. How is the result interpreted?
  beginner · programming · The Deutsch–Jozsa Algorithm
• Multiple choicecorenew
  For deciding whether an n-bit function is constant or balanced (with certainty), what is the worst-case classical query complexity, while Deutsch-Jozsa uses exactly one query?
  beginner · programming · Quantum vs Classical Queries
• Multiple choicecorenew
  For the n = 2 Deutsch-Jozsa problem (decide constant vs balanced with certainty), how many oracle queries does a classical algorithm need in the worst case, versus the quantum algorithm?
  beginner · programming · Checkpoint: Solve It in One Query
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |0⟩?
  beginner · programming · Project: A Quantum Coin Flip
• Numericcorenew
  After running this circuit, what is the expectation value ⟨X⟩ of qubit 0 (its Bloch X-component)?
  beginner · programming · Project: A Quantum Coin Flip
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 6369). Predict how many give |0⟩. Estimate P(|0⟩) × 1024 and pick the closest option.
  beginner · programming · Project: A Quantum Coin Flip
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|0⟩)?
  beginner · programming · Project: A Biased Coin
• Numericcorenew
  This circuit prepares qubit 0 in some state. What expectation value ⟨X⟩ does it have along the Bloch X-axis?
  beginner · programming · Project: A Biased Coin
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 2925). How many of the shots come out |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  beginner · programming · Project: A Biased Coin
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |0000⟩?
  beginner · programming · Project: A Quantum Random Number
• Numericcorenew
  For qubit 0 after this circuit, what is ⟨X⟩, the X-axis projection of its Bloch vector?
  beginner · programming · Project: A Quantum Random Number
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 5958), how many shots land on |0000⟩? Estimate P(|0000⟩) × 1024 and pick the closest option.
  beginner · programming · Project: A Quantum Random Number
• Numericcorenew
  After running this circuit, what is the probability that all 4 qubits together read |0001⟩ (qubit 0 leftmost)?
  beginner · programming · Project: A Quantum Random Number
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |00⟩?
  beginner · programming · Project: Two-Qubit Correlations
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 2995). How many of the shots come out |00⟩? Estimate P(|00⟩) × 1024 and pick the closest option.
  beginner · programming · Project: Two-Qubit Correlations
• Numericcorenew
  After this circuit, what is P(|11⟩) for the whole 2-qubit register measured together (qubit 0 leftmost)?
  beginner · programming · Project: Two-Qubit Correlations
• Numericcorenew
  After running this circuit, what is the probability of measuring |00⟩?
  beginner · programming · Project: A Simple Bell Test
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 9935), how many shots land on |00⟩? Estimate P(|00⟩) × 1024 and pick the closest option.
  beginner · programming · Project: A Simple Bell Test
• Numericcorenew
  For this circuit, what is the probability that measuring all 2 qubits yields exactly |11⟩ (qubit 0 leftmost)?
  beginner · programming · Project: A Simple Bell Test
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |1⟩?
  beginner · programming · Project: Guess the Hidden Gate
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|0⟩)?
  beginner · programming · Project: State Tomography (Intro)
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |000⟩?
  beginner · programming · Project: Build a Mini Circuit Library
• Numericcorenew
  After running this circuit, what is the expectation value ⟨X⟩ of qubit 0 (its Bloch X-component)?
  beginner · programming · Project: Build a Mini Circuit Library
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 836). Predict how many give |000⟩. Estimate P(|000⟩) × 1024 and pick the closest option.
  beginner · programming · Project: Build a Mini Circuit Library
• Numericcorenew
  For this circuit, what is the probability that measuring all 3 qubits yields exactly |001⟩ (qubit 0 leftmost)?
  beginner · programming · Project: Build a Mini Circuit Library
• True / Falsecorenew
  The gates H and T together are universal for single-qubit unitaries: they generate a dense subgroup of SU(2), so any single-qubit operation can be approximated to any desired precision using only those two gates.
  beginner · programming · Reflection: What You Can Now Build
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |000⟩?
  beginner · programming · Capstone: Design Your Own Demo
• Predictcorenew
  On the seeded simulator (seed 6902), this circuit runs for 1024 shots. About how many come out |000⟩? Estimate P(|000⟩) × 1024 and pick the closest option.
  beginner · programming · Capstone: Design Your Own Demo
• Numericcorenew
  Run this circuit, then measure all 3 qubits at once. What is the probability of the joint outcome |111⟩ (qubit 0 leftmost)?
  beginner · programming · Capstone: Design Your Own Demo
• True / Falsecorenew
  In a complex vector space, both vector addition and scalar multiplication must keep the result inside the space V (the closure requirement).
  intermediate · physics · Vector Spaces over ℂ
• Multiple choicecorenew
  Let $|v\rangle = (2+i)|0\rangle + (1-2i)|1\rangle$. What is the squared norm $\langle v|v\rangle = |2+i|^2 + |1-2i|^2$?
  intermediate · physics · Inner Product Spaces
• Multiple choicecorenew
  A Hilbert space is defined as a complex inner-product space with which additional property?
  intermediate · physics · Hilbert Spaces
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |0⟩?
  intermediate · physics · Dirac Bra-Ket Notation
• Numericcorenew
  For qubit 0 after this circuit, what is ⟨X⟩, the X-axis projection of its Bloch vector?
  intermediate · physics · Dirac Bra-Ket Notation
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 3431), how many shots land on |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  intermediate · physics · Dirac Bra-Ket Notation
• Multiple choicecorenew
  Under the ket-to-bra correspondence, if a ket is scaled as $|\psi\rangle \to \alpha|\psi\rangle$ with $\alpha$ a complex number, how does its dual bra transform?
  intermediate · physics · Dual Vectors
• Multiple choicecorenew
  A normalized state is expanded in an orthonormal basis as $c_0|0\rangle + c_1|1\rangle$ with $c_0 = (1+i)/\sqrt{5}$ and $c_1 = \sqrt{3}/\sqrt{5}$. By completeness, what is $|c_0|^2 + |c_1|^2$?
  intermediate · physics · Orthonormal Bases and Completeness
• Multiple choicecorenew
  For the projector $P_0 = |0\rangle\langle 0|$ and the state $|\psi\rangle = \cos(\theta/2)|0\rangle + \sin(\theta/2)|1\rangle$ with $\theta = \pi/3$, what is the expectation $\langle\psi|P_0|\psi\rangle$?
  intermediate · physics · Outer Products and Projectors
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |1⟩?
  intermediate · physics · Representing States in a Basis
• Numericcorenew
  For qubit 0 after this circuit, what is ⟨X⟩, the X-axis projection of its Bloch vector?
  intermediate · physics · Representing States in a Basis
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 1272), how many shots land on |1⟩? Estimate P(|1⟩) × 1024 and pick the closest option.
  intermediate · physics · Representing States in a Basis
• Multiple choicecorenew
  In the continuous position basis, a particle's wavefunction $\psi(x)$ is identified with which of the following?
  intermediate · physics · Continuous Bases (Position)
• Numericcorenew
  After running this circuit, what is the probability of measuring |00⟩?
  intermediate · physics · Tensor-Product Spaces
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 9838), how many shots land on |00⟩? Estimate P(|00⟩) × 1024 and pick the closest option.
  intermediate · physics · Tensor-Product Spaces
• Numericcorenew
  For this circuit, what is the probability that measuring all 2 qubits yields exactly |01⟩ (qubit 0 leftmost)?
  intermediate · physics · Tensor-Product Spaces
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |00⟩?
  intermediate · physics · Checkpoint: Formalism Workout
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 2898). How many of the shots come out |00⟩? Estimate P(|00⟩) × 1024 and pick the closest option.
  intermediate · physics · Checkpoint: Formalism Workout
• Numericcorenew
  After this circuit, what is P(|11⟩) for the whole 2-qubit register measured together (qubit 0 leftmost)?
  intermediate · physics · Checkpoint: Formalism Workout
• Multiple choicecorenew
  An operator $A$ is linear when it satisfies $A(\alpha|\psi\rangle + \beta|\phi\rangle) = \alpha A|\psi\rangle + \beta A|\phi\rangle$. The operation 'take the complex conjugate of every amplitude' is best described as which of the following?
  intermediate · physics · Linear Operators
• Multiple choicecorenew
  The Pauli-Y operator acts as $Y|0\rangle = i|1\rangle$ and $Y|1\rangle = -i|0\rangle$. What is the imaginary part of the matrix element $\langle 1|Y|0\rangle$?
  intermediate · physics · Matrix Elements
• Multiple choicecorenew
  A 2x2 matrix A has rows (1, 2i) and (3, 4). For the adjoint A-dagger, where (A-dagger)[i][j] = conj(A[j][i]), what is the imaginary part of the entry in row 1, column 0?
  intermediate · physics · Adjoint Operators
• True / Falsecorenew
  Consider the matrix A with rows (2, 1-i) and (1+i, 3). A is Hermitian if A[i][j] = conj(A[j][i]) for all entries. Is A Hermitian?
  intermediate · physics · Hermitian Operators
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |0⟩?
  intermediate · physics · Unitary Operators
• Numericcorenew
  For qubit 0 after this circuit, what is ⟨X⟩, the X-axis projection of its Bloch vector?
  intermediate · physics · Unitary Operators
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 3235), how many shots land on |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  intermediate · physics · Unitary Operators
• Multiple choicecorenew
  Find the larger eigenvalue of the Hermitian matrix $A = \begin{bmatrix}3 & 1 \\ 1 & 3\end{bmatrix}$. (Hint: eigenvalues are $(\mathrm{tr} \pm \sqrt{\mathrm{tr}^2 - 4\det})/2$ with $\mathrm{tr} = 6$, $\det = 8$.)
  intermediate · physics · Eigenvalue Equations
• Multiple choicecorenew
  According to the spectral theorem for a Hermitian operator A on a finite-dimensional complex Hilbert space, which statement is guaranteed to be true?
  intermediate · physics · The Spectral Theorem
• Multiple choicecorenew
  The matrix $A = \begin{bmatrix}3 & 1 \\ 1 & 3\end{bmatrix}$ has eigenvalues 4 and 2. For the function $f(x) = \sqrt{x}$, what is $\mathrm{Tr}(f(A)) = \sqrt{4} + \sqrt{2}$?
  intermediate · physics · Functions of Operators
• Multiple choicecorenew
  For a projector P (an orthogonal projector) used in a projection-valued measure, which pair of defining properties holds?
  intermediate · physics · Projection-Valued Measures
• Multiple choicecorenew
  With Pauli matrices $X = \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}$ and $Z = \begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}$, compute the commutator $[X,Z] = XZ - ZX$. What is its (0,1) entry?
  intermediate · physics · Commuting Operators and Common Eigenbases
• Multiple choicecorenew
  The observable $A = \begin{bmatrix}1 & 2 \\ 2 & 1\end{bmatrix}$ has eigenvalues 3 and -1 with normalized eigenvectors $(1,1)/\sqrt{2}$ and $(1,-1)/\sqrt{2}$. Using the spectral decomposition $A = \sum_k \lambda_k |v_k\rangle\langle v_k|$, what is the reconstructed (0,0) entry?
  intermediate · physics · Checkpoint: Diagonalize an Observable
• Multiple choicecorenew
  Postulate 1 says a quantum system is described by a unit state vector. Two state vectors $|\psi\rangle$ and $e^{i\gamma}|\psi\rangle$ that differ only by an overall phase represent what?
  intermediate · physics · Postulate 1: State Vectors
• Multiple choicecorenew
  By Postulate 2, the possible results of measuring an observable are the eigenvalues of its associated operator. Why must that operator be Hermitian?
  intermediate · physics · Postulate 2: Observables
• Multiple choicecorenew
  By Postulate 3, measuring observable $A = \sum_k a_k P_k$ on a normalized state $|\psi\rangle$ yields outcome $a_k$ with what probability?
  intermediate · physics · Postulate 3: Measurement
• Multiple choicecorenew
  For $|\psi\rangle = \cos(\theta)|0\rangle + \sin(\theta)|1\rangle$ with $\theta = \pi/6$, and $|+\rangle = (|0\rangle + |1\rangle)/\sqrt{2}$, what is $p(+) = |\langle+|\psi\rangle|^2 = (1/2)(\cos(\theta) + \sin(\theta))^2$?
  intermediate · physics · The Born Rule, Formally
• Multiple choicecorenew
  A two-qubit state is $(1/\sqrt{10})(|00\rangle + 2|01\rangle + 2|10\rangle + |11\rangle)$, with amplitudes labeled $c_{q0 q1}$. Qubit 0 is measured and gives 0. In the collapsed (renormalized) state, what is the probability that qubit 1 = 1?
  intermediate · physics · State Update After Measurement
• Multiple choicecorenew
  Postulate 4 states that a closed quantum system evolves by a unitary operator U with U^dagger U = I. What key property does this guarantee for the state?
  intermediate · physics · Postulate 4: Time Evolution
• Multiple choicecorenew
  By Postulate 5, the state space of a composite system is the tensor product of the component spaces. If system A has dimension d_A and system B has dimension d_B, what is the dimension of the joint space?
  intermediate · physics · Postulate 5: Composite Systems
• Multiple choicecorenew
  For $|\psi\rangle = \cos(\theta)|0\rangle + \sin(\theta)|1\rangle$ with $\theta = \pi/6$, the Z eigenvalues are +1 for $|0\rangle$ and -1 for $|1\rangle$. What is the expectation $\langle Z\rangle = \cos^2(\theta) - \sin^2(\theta)$?
  intermediate · physics · Expectation Values, Formally
• Multiple choicecorenew
  Two Hermitian operators A and B possess a complete set of simultaneous (common) eigenvectors if and only if which condition holds?
  intermediate · physics · Compatible and Incompatible Observables
• Multiple choicecorenew
  Sum four results from the measurement postulates: (A) for $|\psi\rangle = \cos(\theta)|0\rangle + \sin(\theta)|1\rangle$ with $\theta = \pi/6$, $p(1) = \sin^2(\theta)$; (B) $\langle Z\rangle = \cos^2(\theta) - \sin^2(\theta)$ for the same state; (C) for $(1/\sqrt{10})(|00\rangle + 2|01\rangle + ...)$, after measuring q0 = 0 the conditional prob that q1 = 1; (D) $\langle Z\otimes Z\rangle$ for $(1/\sqrt{2})(|00\rangle + |11\rangle)$. What is A + B + C + D?
  intermediate · physics · Checkpoint: Apply the Postulates
• True / Falsecorenew
  The closed-form time-evolution operator U(t) = e^(-iHt/hbar) is valid for any Hamiltonian, including ones that depend on time.
  intermediate · physics · The Time-Evolution Operator
• Multiple choicecorenew
  Solving the Schrödinger equation, the survival probability of staying in the initial state is $P = \cos^2(\omega t/2)$. With $\omega t = \pi/3$, what is $P$?
  intermediate · physics · Solving the Schrödinger Equation
• Multiple choicecorenew
  An energy eigenstate evolves as $|\psi(t)\rangle = e^{-i E t/\hbar}|E\rangle$, which here gives amplitudes $a_{re} = \cos(\omega t/2)$ and $a_{im} = \sin(\omega t/2)$ for a single basis component, with $\omega t = 1.234$. What is the total probability $a_{re}^2 + a_{im}^2$?
  intermediate · physics · Energy Eigenstates Evolve Simply
• Multiple choicecorenew
  A superposition of two energy levels with gap $\omega_{10} = 2$ ($\hbar = 1$) beats with survival probability $\cos^2(\omega_{10} \cdot t / 2)$. At $t = \pi/4$, what is this probability?
  intermediate · physics · Superposition and Beats
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|0⟩)?
  intermediate · physics · Two-Level Rabi Oscillations
• Numericcorenew
  This circuit prepares qubit 0 in some state. What expectation value ⟨Y⟩ does it have along the Bloch Y-axis?
  intermediate · physics · Two-Level Rabi Oscillations
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 4965). How many of the shots come out |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  intermediate · physics · Two-Level Rabi Oscillations
• Multiple choicecorenew
  For a two-level system, the expectation value <X>(t) oscillates as cos(wt). At time t with wt = pi/3, what is cos(wt)?
  intermediate · physics · Expectation Value Dynamics
• Multiple choicecorenew
  Ehrenfest's theorem for the average position and momentum of a particle gives d/dt <p> = -<dV/dx>. Why is this not exactly Newton's law d/dt p = -V'(<x>)?
  intermediate · physics · Ehrenfest’s Theorem
• Multiple choicecorenew
  For a Hamiltonian H = (w/2) Z, the observable Z is conserved if it commutes with H, i.e. [H, Z] = HZ - ZH = 0. What is the (0,0) entry of HZ - ZH?
  intermediate · physics · Conserved Quantities
• Multiple choicecorenew
  In the Heisenberg picture, the state is held fixed at $|\psi(0)\rangle$ while operators carry the time dependence. How does an operator evolve?
  intermediate · physics · The Heisenberg Picture (Preview)
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|0⟩)?
  intermediate · physics · Spin Precession Revisited
• Numericcorenew
  This circuit prepares qubit 0 in some state. What expectation value ⟨Y⟩ does it have along the Bloch Y-axis?
  intermediate · physics · Spin Precession Revisited
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 3862). How many of the shots come out |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  intermediate · physics · Spin Precession Revisited
• Multiple choicecorenew
  Under Larmor-type precession the Bloch vector traces <X>(t) = cos(wt), <Y>(t) = sin(wt), <Z>(t) = 0. At wt = 2*pi/3, what is <X>(t)?
  intermediate · physics · Checkpoint: Evolve a State
• Multiple choicecorenew
  For the quantum harmonic oscillator, the energy levels are E_n = hbar*omega*(n + 1/2) for n = 0, 1, 2, .... What is the energy spacing between adjacent levels?
  intermediate · physics · Recap of the Oscillator
• Multiple choicecorenew
  The harmonic-oscillator Hamiltonian is $H = a^\dagger a + 1/2$ (in units of $\hbar\omega$). What is the ground-state energy $E_0 = \langle 0| H |0\rangle$?
  intermediate · physics · Creation and Annihilation Operators
• Multiple choicecorenew
  Using the number-basis matrices for $a$ and $a^\dagger$ (with $a|n\rangle = \sqrt{n}|n-1\rangle$, $a^\dagger|n\rangle = \sqrt{n+1}|n+1\rangle$), evaluate the diagonal element of the commutator $[a, a^\dagger] = a a^\dagger - a^\dagger a$ on the state $|1\rangle$, i.e. $\langle 1| [a,a^\dagger] |1\rangle$.
  intermediate · physics · The Commutator [a, a†]
• Multiple choicecorenew
  The number operator is $N = a^\dagger a$, with $N|n\rangle = n|n\rangle$. Summing its diagonal entries over the first three levels, what is $\langle 0|N|0\rangle + \langle 1|N|1\rangle + \langle 2|N|2\rangle$?
  intermediate · physics · The Number Operator
• Multiple choicecorenew
  Starting from the ground state $|0\rangle$, the raising operator is applied three times: $a^\dagger a^\dagger a^\dagger |0\rangle$. Using $a^\dagger|n\rangle = \sqrt{n+1}|n+1\rangle$, what is the accumulated coefficient (the norm of the resulting unnormalized state)?
  intermediate · physics · Raising and Lowering States
• Multiple choicecorenew
  Harmonic-oscillator energies are E_n = hbar*omega*(n + 1/2). In units where hbar*omega = 1, what is E_4 - E_2?
  intermediate · physics · Energy Spectrum from Ladders
• Multiple choicecorenew
  For the harmonic oscillator with $x = (a + a^\dagger)/\sqrt{2}$ (natural units), what is the ground-state expectation $\langle 0| x^2 |0\rangle$?
  intermediate · physics · The Ground State
• Multiple choicecorenew
  With $x = (a + a^\dagger)/\sqrt{2}$ for the harmonic oscillator (natural units), what is the expectation $\langle 1| x^2 |1\rangle$ in the first excited state?
  intermediate · physics · Position and Momentum in Terms of a, a†
• Multiple choicecorenew
  A coherent state $|\alpha\rangle$ of the harmonic oscillator is defined by which property?
  intermediate · physics · Coherent States (Intro)
• Multiple choicecorenew
  With $x = (a + a^\dagger)/\sqrt{2}$ for the harmonic oscillator (natural units), compute the matrix element $\langle 2| x |3\rangle$. (Recall $a|n\rangle = \sqrt{n}|n-1\rangle$.)
  intermediate · physics · Matrix Elements of x and p
• Multiple choicecorenew
  For the quantum harmonic oscillator with $x = (a + a^\dagger)/\sqrt{2}$ in natural units, compute the matrix element $\langle 2| x^2 |0\rangle$. (Use $a|n\rangle = \sqrt{n}|n-1\rangle$ and $a^\dagger|n\rangle = \sqrt{n+1}|n+1\rangle$.)
  intermediate · physics · Checkpoint: Ladder Algebra
• True / Falsecorenew
  In quantum mechanics, the orbital angular momentum component operators L_x, L_y, and L_z are each Hermitian, which is why orbital angular momentum is an observable with real measured values.
  intermediate · physics · Angular Momentum Operators
• Multiple choicecorenew
  For spin-1/2 operators S_x, S_y, S_z = (1/2)*sigma_x, (1/2)*sigma_y, (1/2)*sigma_z (hbar = 1), the commutator satisfies [S_x, S_y] = i*c*S_z. What is the coefficient c?
  intermediate · physics · Commutation Relations
• Multiple choicecorenew
  For a spin-1/2 system, $S^2 = S_x^2 + S_y^2 + S_z^2$. What is the value of the commutator $[S^2, S_z]$ (reported as the total magnitude of its matrix entries)?
  intermediate · physics · The Total Angular Momentum L²
• Multiple choicecorenew
  The raising operator acts as $L_+ |l,m\rangle = \sqrt{l(l+1) - m(m+1)} |l,m+1\rangle$. For $l = 1/2$ and $m = -1/2$, what is the coefficient $\sqrt{l(l+1) - m(m+1)}$?
  intermediate · physics · Raising and Lowering Operators L±
• Multiple choicecorenew
  For an orbital angular momentum state with quantum number $l = 2$, the operator $L^2$ has eigenvalue $l(l+1)$ (in units where $\hbar = 1$). What is that eigenvalue?
  intermediate · physics · Eigenvalues of L² and L_z
• Multiple choicecorenew
  For a given orbital quantum number l = 3, the magnetic quantum number m takes integer values from -l to +l. How many distinct values of m are there?
  intermediate · physics · Quantum Numbers ℓ and m
• Multiple choicecorenew
  The spherical harmonic $Y_\ell^m(\theta, \phi)$ is a simultaneous eigenfunction of $L^2$ and $L_z$. What are its eigenvalues?
  intermediate · physics · Spherical Harmonics
• Multiple choicecorenew
  Orbital angular momentum (quantum number ell) and spin angular momentum (quantum number s) obey the same algebra but differ in allowed values. Which statement is correct?
  intermediate · physics · Orbital vs Spin Angular Momentum
• Multiple choicecorenew
  For $l = 1$ with basis ordering $m = +1, 0, -1$, the raising operator acts as $L_+|l,m\rangle = \sqrt{l(l+1) - m(m+1)}|l,m+1\rangle$. What is the matrix element $\langle l,0| L_+ |l,-1\rangle$?
  intermediate · physics · Matrix Representations
• Multiple choicecorenew
  In the vector model, an angular momentum state has length $|L| = \hbar\sqrt{\ell(\ell+1)}$ and maximum z-projection $\hbar\ell$ (the $m = \ell$ state). Why can the vector never point exactly along the z-axis?
  intermediate · physics · Vector Model of Angular Momentum
• Multiple choicecorenew
  A spin-1/2 starts in $|0\rangle$ and is rotated about the y-axis to the state $\cos(\theta/2)|0\rangle + \sin(\theta/2)|1\rangle$ with $\theta = \pi/3$. The expectation value $\langle S_z\rangle = (1/2)(p_0 - p_1) = (1/2)\cos(\theta)$, where $p_0=\cos^2(\theta/2)$ and $p_1=\sin^2(\theta/2)$. What is $\langle S_z\rangle$ (in units of $\hbar$)?
  intermediate · physics · Expectation Values of Components
• Multiple choicecorenew
  The lowering operator acts as $L_-|l,m\rangle = \sqrt{l(l+1) - m(m-1)}|l,m-1\rangle$. For $l = 3/2$ lowering the state $m = +1/2$, what is the coefficient?
  intermediate · physics · Checkpoint: Angular Momentum Algebra
• Numericcorenew
  After running this circuit, what is the probability of measuring |0⟩?
  intermediate · physics · Spin-½ as a Two-Level System
• Numericcorenew
  Run this circuit. What is qubit 0's Bloch X-component — the expectation value ⟨X⟩?
  intermediate · physics · Spin-½ as a Two-Level System
• Predictcorenew
  On the seeded simulator (seed 2387), this circuit runs for 1024 shots. About how many come out |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  intermediate · physics · Spin-½ as a Two-Level System
• Multiple choicecorenew
  Let $\sigma_y = \begin{bmatrix}0 & -i \\ i & 0\end{bmatrix}$. Compute the matrix product $\sigma_y \cdot \sigma_y$. What is the real part of its top-left (0,0) entry?
  intermediate · physics · Pauli Matrices Revisited
• Multiple choicecorenew
  Using the Pauli matrices $\sigma_x = \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}$ and $\sigma_y = \begin{bmatrix}0 & -i \\ i & 0\end{bmatrix}$, compute the commutator $[\sigma_x, \sigma_y] = \sigma_x \sigma_y - \sigma_y \sigma_x$. What is the imaginary part of its top-left (0,0) entry?
  intermediate · physics · Pauli Algebra and Anticommutators
• Multiple choicecorenew
  The spin operator $S_z = (\hbar/2) \sigma_z$, where $\sigma_z$ has eigenvalues $+1$ (for $|0\rangle$) and $-1$ (for $|1\rangle$). With $\hbar = 1$, what is the $S_z$ eigenvalue of the spin-up state $|0\rangle$?
  intermediate · physics · Spin Operators Sᵢ = (ħ/2)σᵢ
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |0⟩?
  intermediate · physics · Eigenstates Along Any Axis
• Numericcorenew
  After running this circuit, what is the expectation value ⟨X⟩ of qubit 0 (its Bloch X-component)?
  intermediate · physics · Eigenstates Along Any Axis
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 6635). Predict how many give |0⟩. Estimate P(|0⟩) × 1024 and pick the closest option.
  intermediate · physics · Eigenstates Along Any Axis
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|0⟩)?
  intermediate · physics · Expectation Values of Spin
• Numericcorenew
  This circuit prepares qubit 0 in some state. What expectation value ⟨X⟩ does it have along the Bloch X-axis?
  intermediate · physics · Expectation Values of Spin
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 9668). How many of the shots come out |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  intermediate · physics · Expectation Values of Spin
• Multiple choicecorenew
  A spin in a magnetic field precesses at the Larmor frequency omega_L = gamma * B. For gyromagnetic ratio gamma = 5 and field strength B = 0.4 (hbar = 1), what is omega_L (which also equals the energy splitting Delta_E)?
  intermediate · physics · Spin in a Magnetic Field
• Numericcorenew
  After running this circuit, what is the probability of measuring |0⟩?
  intermediate · physics · Rabi Oscillations of Spin
• Numericcorenew
  Run this circuit. What is qubit 0's Bloch Y-component — the expectation value ⟨Y⟩?
  intermediate · physics · Rabi Oscillations of Spin
• Predictcorenew
  On the seeded simulator (seed 5761), this circuit runs for 1024 shots. About how many come out |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  intermediate · physics · Rabi Oscillations of Spin
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |0⟩?
  intermediate · physics · The Rotation Operator for Spin
• Numericcorenew
  After running this circuit, what is the expectation value ⟨Y⟩ of qubit 0 (its Bloch Y-component)?
  intermediate · physics · The Rotation Operator for Spin
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 8794). Predict how many give |0⟩. Estimate P(|0⟩) × 1024 and pick the closest option.
  intermediate · physics · The Rotation Operator for Spin
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|01⟩)?
  intermediate · physics · Two-Spin Systems
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 1854). Predict how many give |01⟩. Estimate P(|01⟩) × 1024 and pick the closest option.
  intermediate · physics · Two-Spin Systems
• Numericcorenew
  After running this circuit, what is the probability that all 2 qubits together read |10⟩ (qubit 0 leftmost)?
  intermediate · physics · Two-Spin Systems
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |0⟩?
  intermediate · physics · Checkpoint: Spin Dynamics
• Numericcorenew
  For qubit 0 after this circuit, what is ⟨X⟩, the X-axis projection of its Bloch vector?
  intermediate · physics · Checkpoint: Spin Dynamics
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 4887), how many shots land on |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  intermediate · physics · Checkpoint: Spin Dynamics
• Multiple choicecorenew
  When two angular momenta with quantum numbers j1 and j2 are combined, which values of the total quantum number j are allowed?
  intermediate · physics · Combining Two Angular Momenta
• Multiple choicecorenew
  Two angular momenta j1 and j2 are combined. Which set of four mutually commuting operators defines the COUPLED basis |j, m>?
  intermediate · physics · The Coupled and Uncoupled Bases
• Multiple choicecorenew
  Two spin-1/2 qubits are in the state $|up,up\rangle$. Using $S^2/\hbar^2 = S_1^2 + S_2^2 + 2(S_1 \cdot S_2)$, where each spin gives $S^2 = 3/4$, the $S_{1z} S_{2z}$ term gives $(1/2)(1/2)$, and the ladder terms annihilate $|up,up\rangle$, what is $S^2$ (in units of $\hbar^2$)?
  intermediate · physics · Total Spin of Two Qubits
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |01⟩?
  intermediate · physics · Singlet and Triplet States
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 537). How many of the shots come out |01⟩? Estimate P(|01⟩) × 1024 and pick the closest option.
  intermediate · physics · Singlet and Triplet States
• Numericcorenew
  After this circuit, what is P(|10⟩) for the whole 2-qubit register measured together (qubit 0 leftmost)?
  intermediate · physics · Singlet and Triplet States
• Multiple choicecorenew
  Applying the lowering operator to $|1,1\rangle$ gives $S_- |1,1\rangle = \sqrt{j(j+1) - m(m-1)} |1,0\rangle$ with $j=1$, $m=1$, so the normalization is $N = \sqrt{2}$. The state $|1,0\rangle = (1/N)(|down,up\rangle + |up,down\rangle)$. What is the Clebsch-Gordan coefficient of $|up,down\rangle$ in $|1,0\rangle$?
  intermediate · physics · Clebsch–Gordan Coefficients
• Multiple choicecorenew
  Two angular momenta with $j_1 = 1$ and $j_2 = 1/2$ are coupled. The allowed total-J values run from $|j_1 - j_2|$ to $j_1 + j_2$ in integer steps, and the combined space has dimension equal to the sum of $(2J+1)$ over those J. What is that total dimension?
  intermediate · physics · Allowed Total-J Values
• Multiple choicecorenew
  An electron in a p-state (orbital ell = 1, spin s = 1/2) couples L and S into a total J. What are the allowed total angular momentum quantum numbers j?
  intermediate · physics · Adding Spin and Orbital Momentum
• Multiple choicecorenew
  Using J = L + S, the spin-orbit operator L.S can be rewritten so that on a coupled state |j, m_j> its expectation value depends only on j, ell, and s. Which identity gives this?
  intermediate · physics · Fine-Structure Preview
• Multiple choicecorenew
  For two spin-1/2 particles, how do the coupled spin states behave under exchange (the swap operator P_12)?
  intermediate · physics · Symmetry of Combined States
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |01⟩?
  intermediate · physics · Checkpoint: Add Two Spins
• Predictcorenew
  On the seeded simulator (seed 8762), this circuit runs for 1024 shots. About how many come out |01⟩? Estimate P(|01⟩) × 1024 and pick the closest option.
  intermediate · physics · Checkpoint: Add Two Spins
• Numericcorenew
  Run this circuit, then measure all 2 qubits at once. What is the probability of the joint outcome |10⟩ (qubit 0 leftmost)?
  intermediate · physics · Checkpoint: Add Two Spins
• Multiple choicecorenew
  In the hydrogen atom, the Coulomb potential energy between the proton $(+e)$ and electron $(-e)$ at separation $r$ is $V(r) = -e^2 / (4\pi\epsilon_0 r)$. Which feature of this potential is the structural fact that makes the atom exactly solvable?
  intermediate · physics · The Coulomb Potential
• Multiple choicecorenew
  Separating the hydrogen wavefunction as $\psi = R(r)Y_\ell^m(\theta, \phi)$ reduces the 3D Schrodinger equation to a 1D radial equation. In that radial equation, what is the term $\hbar^2\ell(\ell+1)/(2\mu r^2)$?
  intermediate · physics · Separating Radial and Angular Parts
• Multiple choicecorenew
  For the hydrogen radial equation written with the reduced radial function $u(r) = r R(r)$, the effective potential is $V_{eff}(r) = -e^2/(4\pi\epsilon_0 r) + \hbar^2 \ell(\ell+1)/(2\mu r^2)$. What is the role of the second term?
  intermediate · physics · The Radial Equation
• Multiple choicecorenew
  An electron in hydrogen falls from $n=3$ to $n=2$, where $E_n = -13.6$ eV $/ n^2$. The emitted photon's energy equals $E_3 - E_2$ (the energy lost). What is that photon energy in eV?
  intermediate · physics · Energy Levels Revisited
• Multiple choicecorenew
  For a hydrogen radial wavefunction R_{n,ell}(r), the number of radial nodes (zeros at finite r > 0) is given by which expression?
  intermediate · physics · Radial Wavefunctions
• Multiple choicecorenew
  For the hydrogen shell $n = 3$, the orbital degeneracy is the sum of $(2l+1)$ over $l = 0,1,2$, and each orbital state holds 2 spin states. What is the total number of states (including spin)?
  intermediate · physics · Degeneracy and Quantum Numbers
• Multiple choicecorenew
  For the hydrogen 1s ground state, the probability density $|\psi|^2$ is largest at $r = 0$, yet the radial distribution function $P(r) = r^2|R(r)|^2$ peaks elsewhere. Where does $P(r)$ peak?
  intermediate · physics · Probability Densities and Orbitals
• Multiple choicecorenew
  For a hydrogen orbital with $n = 2$ and $l = 1$, the expectation value of the radius is $\langle r\rangle/a_0 = (1/2)(3n^2 - l(l+1))$. What is $\langle r\rangle$ in units of the Bohr radius $a_0$?
  intermediate · physics · Expectation of Radius ⟨r⟩
• Multiple choicecorenew
  What are the electric-dipole selection rules for transitions in hydrogen?
  intermediate · physics · Selection Rules for Transitions
• Multiple choicecorenew
  In the normal Zeeman effect, the energy splitting between adjacent magnetic sublevels (Delta m_l = 1) is Delta E = mu_B * B * Delta m_l. With the Bohr magneton mu_B = 5.788e-5 eV/T and a field B = 2 T, what is Delta E in eV?
  intermediate · physics · The Zeeman Effect (Intro)
• Multiple choicecorenew
  The fine structure of hydrogen is smaller than the gross (Bohr) structure by what factor, where $\alpha$ is the fine-structure constant (approximately $1/137$)?
  intermediate · physics · Fine Structure (Overview)
• Multiple choicecorenew
  In hydrogen, $E_n = -13.6$ eV $/ n^2$. The ionization energy from the $n=2$ level is $I = -E_2 = +3.4$ eV, and the $n=2$ level's total degeneracy including spin is $g = 2n^2 = 8$. What is the product $I \cdot g$ (in eV)?
  intermediate · physics · Checkpoint: Hydrogen in Depth
• Multiple choicecorenew
  The indistinguishability requirement for two identical particles states that any physical prediction must be unchanged when the particle labels are swapped, written $|\psi(x_a, x_b)|^2 = |\psi(x_b, x_a)|^2$. What does this condition constrain, and what does it leave free?
  intermediate · physics · Indistinguishability
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |01⟩?
  intermediate · physics · Symmetric and Antisymmetric States
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 6483). How many of the shots come out |01⟩? Estimate P(|01⟩) × 1024 and pick the closest option.
  intermediate · physics · Symmetric and Antisymmetric States
• Numericcorenew
  After this circuit, what is P(|10⟩) for the whole 2-qubit register measured together (qubit 0 leftmost)?
  intermediate · physics · Symmetric and Antisymmetric States
• Multiple choicecorenew
  According to the spin-statistics theorem, what determines whether a fundamental particle is a boson or a fermion?
  intermediate · physics · Bosons and Fermions
• Multiple choicecorenew
  The Pauli exclusion principle is encoded in the antisymmetric two-fermion state $(1/\sqrt{2})(a b - b a)$. If both electrons are placed in the identical single-particle state with $a = b = 0.8$, what is the resulting amplitude?
  intermediate · physics · The Pauli Exclusion Principle
• Multiple choicecorenew
  A two-electron Slater determinant uses spatial amplitudes $a_1 = 0.6$, $a_2 = 0.3$ (orbital 1) and $b_1 = 0.2$, $b_2 = 0.5$ (orbital 2). The antisymmetric amplitude is $(a_1 b_2 - b_1 a_2)/\sqrt{2}$. What value does it take?
  intermediate · physics · Slater Determinants
• True / Falsecorenew
  True or false: For two identical particles placed in distinct orbitals with nonzero exchange overlap, the symmetric spatial state has a SMALLER mean-square separation than the distinguishable case (the particles are drawn together), while the antisymmetric spatial state has a LARGER mean-square separation (the particles are pushed apart).
  intermediate · physics · Exchange Forces
• Multiple choicecorenew
  Two electrons occupy an infinite square well with single-particle energies $E_n = n^2 E_1$ (take $E_1 = 1$). In the ground state, both electrons (opposite spins) fit in $n = 1$. What is the total energy in units of $E_1$?
  intermediate · physics · Two Electrons in a Well
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |01⟩?
  intermediate · physics · Spin and Spatial Symmetry
• Predictcorenew
  On the seeded simulator (seed 4735), this circuit runs for 1024 shots. About how many come out |01⟩? Estimate P(|01⟩) × 1024 and pick the closest option.
  intermediate · physics · Spin and Spatial Symmetry
• Numericcorenew
  Run this circuit, then measure all 2 qubits at once. What is the probability of the joint outcome |10⟩ (qubit 0 leftmost)?
  intermediate · physics · Spin and Spatial Symmetry
• Multiple choicecorenew
  The average occupation of a single-particle level of energy epsilon is n_BE = 1/(e^((epsilon-mu)/kT) - 1) for bosons and n_FD = 1/(e^((epsilon-mu)/kT) + 1) for fermions. What is the only difference between the two distributions, and what does it enforce for fermions?
  intermediate · physics · Bose–Einstein and Fermi–Dirac Statistics
• Multiple choicecorenew
  In simulating fermions on a quantum computer, the Jordan-Wigner transformation maps each fermionic creation/annihilation operator to a qubit operator with a trailing string of which gates, in order to reproduce the fermionic exchange (anticommutation) signs?
  intermediate · physics · Identical Particles in Computing
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |01⟩?
  intermediate · physics · Checkpoint: Exchange Symmetry
• Predictcorenew
  On the seeded simulator (seed 9175), this circuit runs for 1024 shots. About how many come out |01⟩? Estimate P(|01⟩) × 1024 and pick the closest option.
  intermediate · physics · Checkpoint: Exchange Symmetry
• Numericcorenew
  Run this circuit, then measure all 2 qubits at once. What is the probability of the joint outcome |10⟩ (qubit 0 leftmost)?
  intermediate · physics · Checkpoint: Exchange Symmetry
• Multiple choicecorenew
  In time-independent perturbation theory one writes $H = H^{(0)} + \lambda H'$, where $H^{(0)}$ is solvable. The first-order energy shift $E_n^{(1)}$ turns out to equal which quantity?
  intermediate · physics · Why Perturbation Theory
• Multiple choicecorenew
  A qubit in state $|0\rangle$ is perturbed by $H' = 0.3\,\sigma_z + 0.5\,\sigma_x$. Using $\sigma_z|0\rangle = +|0\rangle$ and $\sigma_x|0\rangle = |1\rangle$, the first-order shift is $E^{(1)} = \langle 0|H'|0\rangle$. What is $E^{(1)}$?
  intermediate · physics · First-Order Energy Corrections
• Multiple choicecorenew
  The first-order correction to a state mixes in a neighbor with coefficient $c_1 = \langle 1|H'|0\rangle/(E_0 - E_1)$. With coupling $\langle 1|H'|0\rangle = 0.6$, $E_0 = 1$, and $E_1 = 4$, what is $c_1$?
  intermediate · physics · First-Order State Corrections
• Multiple choicecorenew
  The second-order energy correction from a single coupled level is $E^{(2)} = |V|^2/(E_0 - E_1)$, with $V = \langle 1|H'|0\rangle = 0.6$, $E_0 = 1$, and $E_1 = 4$. What is $E^{(2)}$?
  intermediate · physics · Second-Order Energy Corrections
• Multiple choicecorenew
  Degenerate perturbation theory diagonalizes the 2x2 matrix $\begin{bmatrix}a & b \\ b & d\end{bmatrix}$ with $a = 0, b = 0.2, d = 0$. The eigenvalues are mean $\pm \sqrt{((a-d)/2)^2 + b^2}$. What is the larger ($\lambda_+$) eigenvalue?
  intermediate · physics · Degenerate Perturbation Theory
• Multiple choicecorenew
  In the linear Stark effect, a degenerate pair couples through an off-diagonal element c = -3 (in units of e*a0*E), giving eigenvalues +/-|c|. What is the magnitude of the resulting energy shift?
  intermediate · physics · The Stark Effect
• Multiple choicecorenew
  The Lande g-factor is g_J = 1 + [j(j+1) + s(s+1) - l(l+1)] / [2 j(j+1)]. For a state with l = 1, s = 1/2, j = 3/2, what is g_J?
  intermediate · physics · The Zeeman Effect (Perturbative)
• Multiple choicecorenew
  Fine-structure corrections in hydrogen scale as the square of the fine-structure constant, $\alpha = 1/137.036$. What is the numerical value of $\alpha^2$?
  intermediate · physics · Fine Structure Corrections
• Multiple choicecorenew
  Time-independent perturbation theory is reliable when, for every state m that the perturbation couples to the state n, the ratio |<m|H'|n>| / |E_n - E_m| is much less than 1. Which situation makes the expansion FAIL even when the perturbation H' itself is tiny?
  intermediate · physics · Validity of Perturbation Theory
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |1⟩?
  intermediate · physics · Two-Level Perturbation
• Numericcorenew
  After running this circuit, what is the expectation value ⟨Z⟩ of qubit 0 (its Bloch Z-component)?
  intermediate · physics · Two-Level Perturbation
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 5878). Predict how many give |1⟩. Estimate P(|1⟩) × 1024 and pick the closest option.
  intermediate · physics · Two-Level Perturbation
• Multiple choicecorenew
  For a harmonic oscillator perturbed by $H' = \lambda x^4$, the first-order energy shift is $E^{(1)} = \lambda \langle n|x^4|n\rangle$, where (in the convention used here) $\langle n|x^4|n\rangle = 3(2n^2 + 2n + 1)$. For the ground state $n = 0$ with $\lambda = 1$, what is $E^{(1)}$?
  intermediate · physics · Anharmonic Oscillator
• Multiple choicecorenew
  A state has unperturbed energy $E_0 = 2$ and a perturbation $H'$ with matrix elements $\langle 0|H'|0\rangle = 0.4$ and $\langle 1|H'|0\rangle = 0.3$; the other level sits at $E_1 = 5$. Through second order, $E = E_0 + \langle 0|H'|0\rangle + |\langle 1|H'|0\rangle|^2/(E_0 - E_1)$. What is $E$?
  intermediate · physics · Checkpoint: Perturb a System
• Multiple choicecorenew
  The variational principle states that for any normalizable trial state $\psi$, the Rayleigh quotient $E[\psi] = \langle\psi|H|\psi\rangle / \langle\psi|\psi\rangle$ satisfies which relation with the true ground-state energy $E_0$?
  intermediate · physics · The Variational Principle
• Multiple choicecorenew
  A variational calculation gives the trial energy E(a) = a/2 + 1/(8a) for a positive parameter a. Minimizing over a, what is the minimum energy E(a*)?
  intermediate · physics · Trial Wavefunctions
• Multiple choicecorenew
  A variational ground-state estimate has $E(a) = 1.5\,a - 2\sqrt{2a/\pi}$. Minimizing over the parameter $a$ gives the closed-form result $E_{min} = -4/(3\pi)$. What is $E_{min}$ numerically?
  intermediate · physics · Estimating Ground-State Energy
• Multiple choicecorenew
  In the one-parameter variational treatment of the helium ground state (Z = 2) using a product of hydrogen-like 1s orbitals with effective charge Z_eff, minimizing the energy gives Z_eff = 27/16 is approx 1.69. Why is the optimal Z_eff less than the bare nuclear charge of 2?
  intermediate · physics · Variational Helium (Overview)
• Multiple choicecorenew
  The linear variational method diagonalizes $\begin{bmatrix}a & b \\ b & d\end{bmatrix}$ with $a = 1$, $b = 1$, $d = 3$. The lower eigenvalue is $\text{mean} - \sqrt{((a-d)/2)^2 + b^2}$. What is it?
  intermediate · physics · Linear Variational Method
• Multiple choicecorenew
  In a classically allowed region ($E > V$), the WKB wavefunction has the form $\psi(x) \approx (C/\sqrt{p(x)})\, \exp(\pm (i/\hbar)\int p\, dx)$. What is the physical meaning of the $1/\sqrt{p}$ amplitude factor?
  intermediate · physics · The WKB Approximation
• Multiple choicecorenew
  For a potential giving the classical action $S(E) = \int \sqrt{2E - x^2}\, dx$ between the turning points $x = \pm\sqrt{2E}$, the WKB (Bohr-Sommerfeld) condition is $S(E) = (n + 1/2)\pi$. Solving for the $n = 2$ level, what energy $E$ satisfies $S(E) = (2 + 1/2)\pi$?
  intermediate · physics · WKB Quantization
• Multiple choicecorenew
  A particle of energy $E = 1$ hits a square barrier of height $V_0 = 2$ and width $L = 1$ (units with $\hbar = m = 1$). In the WKB approximation with $|p| = \sqrt{2(V_0-E)}$ constant inside the barrier, the transmission is $T = e^{-2\gamma}$ with $\gamma = L\,|p|$. What is $T$?
  intermediate · physics · Tunneling in WKB
• Multiple choicecorenew
  In the WKB connection formulas, applying the matching at both turning points of a bound state yields the Bohr-Sommerfeld quantization rule integral(p dx) = (n + 1/2) pi hbar. Where does the famous +1/2 come from for two smooth (linear) turning points?
  intermediate · physics · Connection Formulas (Overview)
• Multiple choicecorenew
  For the 1D infinite well ($\hbar = m = 1$, $L = 1$), the variational trial wavefunction $\psi = x(L - x)$ gives an energy $E_{var} = \langle T\rangle/\langle\psi|\psi\rangle$ with $\langle T\rangle = L^3/6$ and $\langle\psi|\psi\rangle = L^5/30$. What is $E_{var}$, and does it correctly upper-bound the exact ground state $E_1 = \pi^2/2 \approx 4.9348$?
  intermediate · physics · Checkpoint: Approximate a Spectrum
• Multiple choicecorenew
  In time-dependent perturbation theory the Hamiltonian is written as $H(t) = H_0 + H'(t)$, and the exact state is expanded as $|\Psi(t)\rangle = \sum_n c_n(t) e^{-i E_n t / \hbar} |n\rangle$. What does $|c_n(t)|^2$ represent?
  intermediate · physics · Time-Dependent Hamiltonians
• True / Falsecorenew
  True or false: In the interaction (Dirac) picture, the equation of motion for the state is i hbar d/dt |Psi_I(t)> = H'_I(t) |Psi_I(t)>, meaning the free Hamiltonian H_0 has completely dropped out of the state's equation of motion, so if the perturbation vanishes the interaction-picture state is constant.
  intermediate · physics · The Interaction Picture
• Multiple choicecorenew
  The first-order transition amplitude magnitude is |c_fi| = (|Vfi|/hbar) * 2|sin(wfi*t/2)| / |wfi|. With hbar = 1, Vfi = 0.5, wfi = 2, and t = 1, what is |c_fi|?
  intermediate · physics · Transition Amplitudes
• Multiple choicecorenew
  A constant perturbation switched on for time $t$ gives transition probability $P = \frac{V_{fi}^2}{\hbar^2} \cdot \frac{4 \sin^2(w_{fi} t/2)}{w_{fi}^2}$. With $\hbar = 1$, $V_{fi} = 0.3$, $w_{fi} = 1.5$, and $t = 2$, what is $P$?
  intermediate · physics · Transition Probabilities
• Multiple choicecorenew
  Under a harmonic perturbation, first-order theory gives the transition probability $P = (V_{fi}^2 / (4\hbar^2)) \cdot \sin^2(\Delta t/2) / (\Delta/2)^2$, where $\Delta = \omega_{fi} - \omega$ is the detuning. With $\hbar = 1$, $V_{fi} = 0.4$, $\omega_{fi} = 5$, $\omega = 4$, and $t = 2$ (so $\Delta = 1$), what is $P$?
  intermediate · physics · Harmonic Perturbations
• Multiple choicecorenew
  Exactly on resonance the transition probability grows quadratically in time: $P = \frac{V_{fi}^2}{4\hbar^2} t^2$. With $\hbar = 1$, $V_{fi} = 0.2$, and $t = 3$, what is $P$?
  intermediate · physics · Resonance
• Multiple choicecorenew
  Fermi's Golden Rule gives the transition rate $W = (2\pi/\hbar)|V_{fi}|^2 \rho$. With $\hbar = 1$, $V_{fi} = 0.1$, and density of final states $\rho = 50$, what is $W$?
  intermediate · physics · Fermi's Golden Rule
• Multiple choicecorenew
  Einstein's detailed-balance argument relates the spontaneous-emission coefficient $A_{ba}$ to the stimulated coefficient $B_{ba}$ via $A_{ba}/B_{ba} = \frac{\hbar \omega_0^3}{\pi^2 c^3}$. What observable consequence follows from the $\omega_0^3$ dependence?
  intermediate · physics · Spontaneous and Stimulated Emission
• Multiple choicecorenew
  For electric-dipole transitions in hydrogen, the angular-momentum selection rules are $\Delta \ell = \pm 1$ and $\Delta m = 0, \pm 1$. Based on these rules, why is the 1s -> 2s transition dipole-forbidden?
  intermediate · physics · Selection Rules from Matrix Elements
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |0⟩?
  intermediate · physics · Driven Two-Level Systems
• Numericcorenew
  For qubit 0 after this circuit, what is ⟨X⟩, the X-axis projection of its Bloch vector?
  intermediate · physics · Driven Two-Level Systems
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 2290), how many shots land on |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  intermediate · physics · Driven Two-Level Systems
• Multiple choicecorenew
  A state decays at the Fermi-golden-rule rate $W = (2\pi/\hbar)|V_{fi}|^2 \rho$ with $\hbar = 1$, $V_{fi} = 0.05$, $\rho = 80$. After exactly one lifetime $\tau = 1/W$, the survival probability is $P = e^{-W\tau}$. What is $P$?
  intermediate · physics · Checkpoint: Transition Rates
• Multiple choicecorenew
  Consider the pure superposition $|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$ versus a 50/50 classical mixture of $|0\rangle$ and $|1\rangle$. When each is measured along the x-axis, what happens?
  intermediate · physics · Pure vs Mixed States
• Multiple choicecorenew
  A pure qubit $|\psi\rangle = \cos(\theta/2)|0\rangle + \sin(\theta/2)|1\rangle$ with $\theta = \pi/3$. Its density operator is $\rho = |\psi\rangle\langle\psi|$. What is the diagonal element $\rho[0][0] = \langle 0|\rho|0\rangle$?
  intermediate · physics · The Density Operator
• True / Falsecorenew
  Consider $\rho = \begin{bmatrix}0.75 & 0.25 \\ 0.25 & 0.25\end{bmatrix}$. A valid density matrix must satisfy: trace $= 1$, Hermitian ($\rho = \rho^\dagger$), and positive semidefinite (all eigenvalues $\geq 0$). Does this $\rho$ satisfy ALL three conditions?
  intermediate · physics · Properties of ρ
• Multiple choicecorenew
  A qubit has density matrix $\rho = \begin{bmatrix}0.75 & 0.25 \\ 0.25 & 0.25\end{bmatrix}$. Its purity is $\mathrm{Tr}(\rho^2)$. What is the purity?
  intermediate · physics · Purity and Tr(ρ²)
• Multiple choicecorenew
  A qubit has density matrix $\rho = \begin{bmatrix}0.75 & 0.25 \\ 0.25 & 0.25\end{bmatrix}$. With $Z = \begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}$, the expectation value is $\langle Z\rangle = \mathrm{Tr}(\rho Z)$. What is $\langle Z\rangle$?
  intermediate · physics · Expectation Values with ρ
• Multiple choicecorenew
  For $\rho = \begin{bmatrix}0.75 & 0.25 \\ 0.25 & 0.25\end{bmatrix}$, the Bloch vector is $r_x = 2\,\mathrm{Re}(\rho_{01})$, $r_y = -2\,\mathrm{Im}(\rho_{01})$, $r_z = \rho_{00} - \rho_{11}$. What is the length $|r| = \sqrt{r_x^2 + r_y^2 + r_z^2}$?
  intermediate · physics · The Bloch Vector of ρ
• Multiple choicecorenew
  The Bell state $|\Phi^+\rangle = (|00\rangle + |11\rangle)/\sqrt{2}$ has reduced density matrix $\rho_A$ obtained by tracing out qubit B. What is the (0,0) entry $\rho_A[0][0]$, i.e. the probability of measuring qubit A in $|0\rangle$?
  intermediate · physics · Reduced Density Matrices
• Multiple choicecorenew
  A two-qubit product state has rho_AB = rho_A (x) rho_B (Kronecker product), with rho_A = diag(0.8, 0.2) and rho_B = diag(0.5, 0.5). After tracing out subsystem B, what is the (0,0) element of the reduced operator rho_A_reduced?
  intermediate · physics · The Partial Trace
• Multiple choicecorenew
  Tracing out one qubit of $\cos(\pi/6)|00\rangle + \sin(\pi/6)|11\rangle$ leaves $\rho_A = \mathrm{diag}(\cos^2(\pi/6), \sin^2(\pi/6))$. The purity is $\mathrm{Tr}(\rho_A^2) = \cos^4(\pi/6) + \sin^4(\pi/6)$. What is it?
  intermediate · physics · Mixed States from Entanglement
• Multiple choicecorenew
  A density matrix $\rho = \begin{bmatrix}0.75 & 0.25 \\ 0.25 & 0.25\end{bmatrix}$ is evolved by the Z gate ($Z = \mathrm{diag}(1,-1)$) as $\rho' = Z \rho Z^\dagger$. What is the off-diagonal element $\rho'[0][1]$?
  intermediate · physics · Time Evolution of ρ
• Multiple choicecorenew
  Two qubits are in the state $\cos(\pi/8)|00\rangle + \sin(\pi/8)|11\rangle$. Tracing out qubit B gives the diagonal reduced density matrix $\rho_A = \mathrm{diag}(\cos^2(\pi/8), \sin^2(\pi/8))$. Compute purity $\mathrm{Tr}(\rho_A^2)$ PLUS $\langle Z\rangle = \mathrm{Tr}(\rho_A Z)$, where $Z = \mathrm{diag}(1, -1)$. What is the sum?
  intermediate · physics · Checkpoint: Work with Density Matrices
• Multiple choicecorenew
  In the EPR argument applied to a spin-singlet pair, EPR concluded that both S_z and S_x of Bob's particle have definite values. Why does this clash with quantum mechanics?
  intermediate · physics · The EPR Paradox
• Multiple choicecorenew
  In a local hidden-variable model, Alice's outcome is written $A(a, \lambda)$ and Bob's as $B(b, \lambda)$. What does the LOCALITY assumption require?
  intermediate · physics · Local Hidden Variables
• True / Falsecorenew
  Bell's theorem states that no local hidden-variable theory can reproduce all the statistical predictions of quantum mechanics.
  intermediate · physics · Bell’s Theorem
• Multiple choicecorenew
  In the CHSH expression S = A*B + A*B' + A'*B - A'*B', each of A, A', B, B' is a deterministic outcome of +1 or -1 (a local hidden-variable model). Over all assignments, what is the maximum possible value of |S|?
  intermediate · physics · The CHSH Inequality
• Multiple choicecorenew
  For the singlet state, the correlation along axes differing by angle (x - y) is E(x, y) = -cos(x - y). With a=0, a'=pi/2, b=pi/4, b'=-pi/4, compute S = E(a,b) + E(a,b') + E(a',b) - E(a',b'). What is |S|?
  intermediate · physics · Quantum Violation of CHSH
• Multiple choicecorenew
  Two qubits are measured along axes separated by angle $\theta = \pi/3$. The correlation is $E = P(\text{same}) - P(\text{opposite})$, with $P(\text{same}) = \sin^2(\theta/2)$ and $P(\text{opposite}) = \cos^2(\theta/2)$. What is $E$?
  intermediate · physics · Measurement Settings and Correlations
• Multiple choicecorenew
  Tsirelson's bound (1980) gives the maximum value of the CHSH quantity |S| achievable by any quantum state and any +/-1-valued local observables. What is that maximum?
  intermediate · physics · Tsirelson’s Bound
• Multiple choicecorenew
  An entanglement witness gives $W = 1/2 - |\langle\Phi^+|\psi\rangle|^2$ for the target Bell state $|\Phi^+\rangle$. The test state is exactly $|\psi\rangle = |\Phi^+\rangle$, so the overlap $|\langle\Phi^+|\psi\rangle|^2 = 1$. What is the witness value $W$?
  intermediate · physics · Entanglement Witnesses
• True / Falsecorenew
  According to the lesson, the first loophole-free Bell tests — closing the locality and detection loopholes simultaneously — were achieved in 2015.
  intermediate · physics · Loopholes in Bell Tests
• Multiple choicecorenew
  For the singlet state $|\Psi^-\rangle$, Bob's reduced density matrix (tracing out Alice's qubit) is $\rho_B = (1/2)$ times the identity. Why does this guarantee no-signalling?
  intermediate · physics · Nonlocality vs Signalling
• Multiple choicecorenew
  For the Bell state |Phi+>, the correlation in the X-Z plane is E(theta_a, theta_b) = cos(theta_a - theta_b). With a=0, a'=pi/2, b=pi/4, b'=-pi/4, evaluate the CHSH quantity S = E(a,b) + E(a,b') + E(a',b) - E(a',b'). What is |S|?
  intermediate · physics · Checkpoint: Violate a Bell Inequality
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|001⟩)?
  intermediate · programming · Registers of n Qubits
• Multiple choicecorenew
  A quantum register holds 3 qubits. How many complex amplitudes are needed to fully describe its pure state (the dimension of its state vector)?
  intermediate · programming · The 2ⁿ State Vector
• Numericcorenew
  After running this circuit, what is the probability of measuring |001⟩?
  intermediate · programming · Tensor Structure of Operations
• Numericcorenew
  Run this circuit. What is qubit 0's Bloch X-component — the expectation value ⟨X⟩?
  intermediate · programming · Tensor Structure of Operations
• Predictcorenew
  On the seeded simulator (seed 3706), this circuit runs for 1024 shots. About how many come out |001⟩? Estimate P(|001⟩) × 1024 and pick the closest option.
  intermediate · programming · Tensor Structure of Operations
• Numericcorenew
  After this circuit, what is P(|101⟩) for the whole 3-qubit register measured together (qubit 0 leftmost)?
  intermediate · programming · Tensor Structure of Operations
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |110⟩?
  intermediate · programming · Big-Endian vs Little-Endian Qubits
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|000⟩)?
  intermediate · programming · Preparing Uniform Superpositions
• Numericcorenew
  This circuit prepares qubit 0 in some state. What expectation value ⟨X⟩ does it have along the Bloch X-axis?
  intermediate · programming · Preparing Uniform Superpositions
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 7613). How many of the shots come out |000⟩? Estimate P(|000⟩) × 1024 and pick the closest option.
  intermediate · programming · Preparing Uniform Superpositions
• Numericcorenew
  Run this circuit, then measure all 3 qubits at once. What is the probability of the joint outcome |001⟩ (qubit 0 leftmost)?
  intermediate · programming · Preparing Uniform Superpositions
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |101⟩?
  intermediate · programming · Computational Basis Encoding
• Multiple choicecorenew
  Per this lesson, the Toffoli gate (CCX) on controls c1, c2 and target t flips the target bit under exactly which condition?
  intermediate · programming · Multi-Controlled Gates
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |1101⟩?
  intermediate · programming · Ancilla Qubits
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|110⟩)?
  intermediate · programming · Uncomputation
• Numericcorenew
  After running this circuit, what is the probability of measuring |010⟩?
  intermediate · programming · Checkpoint: Register Manipulation
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 7080), how many shots land on |010⟩? Estimate P(|010⟩) × 1024 and pick the closest option.
  intermediate · programming · Checkpoint: Register Manipulation
• Numericcorenew
  For this circuit, what is the probability that measuring all 3 qubits yields exactly |101⟩ (qubit 0 leftmost)?
  intermediate · programming · Checkpoint: Register Manipulation
• Multiple choicecorenew
  Per this lesson, in the DFT formula $y_k = (1/\sqrt{N}) \sum_j x_j \cdot \omega_N^{jk}$, what is $\omega_N$?
  intermediate · programming · The Discrete Fourier Transform
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |00⟩?
  intermediate · programming · The QFT Defined
• Numericcorenew
  For qubit 0 after this circuit, what is ⟨X⟩, the X-axis projection of its Bloch vector?
  intermediate · programming · The QFT Defined
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 8200), how many shots land on |00⟩? Estimate P(|00⟩) × 1024 and pick the closest option.
  intermediate · programming · The QFT Defined
• Numericcorenew
  After running this circuit, what is the probability that all 2 qubits together read |01⟩ (qubit 0 leftmost)?
  intermediate · programming · The QFT Defined
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|0⟩)?
  intermediate · programming · QFT on One Qubit
• Numericcorenew
  This circuit prepares qubit 0 in some state. What expectation value ⟨X⟩ does it have along the Bloch X-axis?
  intermediate · programming · QFT on One Qubit
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 5167). How many of the shots come out |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  intermediate · programming · QFT on One Qubit
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |000⟩?
  intermediate · programming · Controlled Rotations in the QFT
• Numericcorenew
  After running this circuit, what is the expectation value ⟨X⟩ of qubit 0 (its Bloch X-component)?
  intermediate · programming · Controlled Rotations in the QFT
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 9948). Predict how many give |000⟩. Estimate P(|000⟩) × 1024 and pick the closest option.
  intermediate · programming · Controlled Rotations in the QFT
• Numericcorenew
  For this circuit, what is the probability that measuring all 3 qubits yields exactly |001⟩ (qubit 0 leftmost)?
  intermediate · programming · Controlled Rotations in the QFT
• Numericcorenew
  After running this circuit, what is the probability of measuring |00⟩?
  intermediate · programming · Building the Two-Qubit QFT
• Numericcorenew
  Run this circuit. What is qubit 0's Bloch X-component — the expectation value ⟨X⟩?
  intermediate · programming · Building the Two-Qubit QFT
• Predictcorenew
  On the seeded simulator (seed 6915), this circuit runs for 1024 shots. About how many come out |00⟩? Estimate P(|00⟩) × 1024 and pick the closest option.
  intermediate · programming · Building the Two-Qubit QFT
• Numericcorenew
  After this circuit, what is P(|01⟩) for the whole 2-qubit register measured together (qubit 0 leftmost)?
  intermediate · programming · Building the Two-Qubit QFT
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |000⟩?
  intermediate · programming · The General QFT Circuit
• Numericcorenew
  For qubit 0 after this circuit, what is ⟨X⟩, the X-axis projection of its Bloch vector?
  intermediate · programming · The General QFT Circuit
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 6041), how many shots land on |000⟩? Estimate P(|000⟩) × 1024 and pick the closest option.
  intermediate · programming · The General QFT Circuit
• Numericcorenew
  After running this circuit, what is the probability that all 3 qubits together read |001⟩ (qubit 0 leftmost)?
  intermediate · programming · The General QFT Circuit
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|000⟩)?
  intermediate · programming · Bit-Reversal and SWAPs
• Numericcorenew
  This circuit prepares qubit 0 in some state. What expectation value ⟨X⟩ does it have along the Bloch X-axis?
  intermediate · programming · Bit-Reversal and SWAPs
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 3008). How many of the shots come out |000⟩? Estimate P(|000⟩) × 1024 and pick the closest option.
  intermediate · programming · Bit-Reversal and SWAPs
• Numericcorenew
  Run this circuit, then measure all 3 qubits at once. What is the probability of the joint outcome |001⟩ (qubit 0 leftmost)?
  intermediate · programming · Bit-Reversal and SWAPs
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |000⟩?
  intermediate · programming · The Inverse QFT
• Numericcorenew
  After running this circuit, what is the expectation value ⟨X⟩ of qubit 0 (its Bloch X-component)?
  intermediate · programming · The Inverse QFT
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 6452). Predict how many give |000⟩. Estimate P(|000⟩) × 1024 and pick the closest option.
  intermediate · programming · The Inverse QFT
• Numericcorenew
  For this circuit, what is the probability that measuring all 3 qubits yields exactly |001⟩ (qubit 0 leftmost)?
  intermediate · programming · The Inverse QFT
• Multiple choicecorenew
  Per this lesson, the QFT circuit on n qubits uses exactly how many two-qubit controlled-phase gates?
  intermediate · programming · QFT vs FFT Complexity
• Numericcorenew
  After running this circuit, what is the probability of measuring |000⟩?
  intermediate · programming · QFT on Periodic States
• Numericcorenew
  Run this circuit. What is qubit 0's Bloch X-component — the expectation value ⟨X⟩?
  intermediate · programming · QFT on Periodic States
• Predictcorenew
  On the seeded simulator (seed 2954), this circuit runs for 1024 shots. About how many come out |000⟩? Estimate P(|000⟩) × 1024 and pick the closest option.
  intermediate · programming · QFT on Periodic States
• Numericcorenew
  After this circuit, what is P(|100⟩) for the whole 3-qubit register measured together (qubit 0 leftmost)?
  intermediate · programming · QFT on Periodic States
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |000⟩?
  intermediate · programming · Checkpoint: Implement the QFT
• Numericcorenew
  After running this circuit, what is the expectation value ⟨X⟩ of qubit 0 (its Bloch X-component)?
  intermediate · programming · Checkpoint: Implement the QFT
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 5987). Predict how many give |000⟩. Estimate P(|000⟩) × 1024 and pick the closest option.
  intermediate · programming · Checkpoint: Implement the QFT
• Numericcorenew
  For this circuit, what is the probability that measuring all 3 qubits yields exactly |001⟩ (qubit 0 leftmost)?
  intermediate · programming · Checkpoint: Implement the QFT
• Multiple choicecorenew
  Per this lesson, why must every eigenvalue $\lambda$ of a unitary operator satisfy $|\lambda| = 1$?
  intermediate · programming · Eigenvalues and Phases
• True / Falsecorenew
  Per this lesson on phase estimation, doubling the precision (one extra bit) costs one extra ancilla qubit and one extra controlled-U call.
  intermediate · programming · The Phase Estimation Problem
• Multiple choicecorenew
  Per this lesson, applying the controlled-S gate (cp(pi/2, 0, 1)) twice in a row implements which controlled operation?
  intermediate · programming · Controlled-U Powers
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|011⟩)?
  intermediate · programming · The QPE Circuit
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |101⟩?
  intermediate · programming · Reading the Phase Register
• Numericcorenew
  After running this circuit, what is the probability of measuring |011⟩?
  intermediate · programming · Precision and Qubit Count
• Multiple choicecorenew
  In the single-qubit QPE of this lesson, after phase kickback the counting qubit holds $(|0\rangle + e^{2\pi i\phi}|1\rangle)/\sqrt{2}$. After the final Hadamard, what are the measurement probabilities?
  intermediate · programming · Single-Qubit Phase Estimation
• Multiple choicecorenew
  For a single Grover marked item in a large database, the success probability after one Grover iteration approaches the limiting value $4/\pi^2$. What is this value numerically?
  intermediate · programming · Success Probability
• Multiple choicecorenew
  Per this lesson, in the quantum-chemistry application of QPE, the time-evolution unitary U = e^(-iHt) is built so that its eigenvalues e^(-i*E_k*t) encode what physical quantities?
  intermediate · programming · Applications of QPE
• Multiple choicecorenew
  In this checkpoint, QPE is run on the S gate with eigenstate $|1\rangle$ (eigenphase $\phi = 1/4$) using a 2-qubit counting register. What bitstring does the counting register yield with certainty?
  intermediate · programming · Checkpoint: Estimate a Phase
• Multiple choicecorenew
  Per this lesson, what is the worst-case classical query complexity of unstructured search over N items, and how does Grover's algorithm compare?
  intermediate · programming · Unstructured Search
• Multiple choicecorenew
  Per this lesson, a phase-flip oracle marking item x* acts as the identity everywhere except for what change on the diagonal entry for |x*>?
  intermediate · programming · The Search Oracle
• Multiple choicecorenew
  Per this lesson's geometric picture of Grover, where the initial state is at angle $\theta$ with $\sin(\theta) = 1/\sqrt{N}$, what does a single Grover iteration do to the state vector?
  intermediate · programming · Amplitude Geometry
• Numericcorenew
  After running this circuit, what is the probability of measuring |00⟩?
  intermediate · programming · The Diffuser (Inversion About Mean)
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 9569), how many shots land on |00⟩? Estimate P(|00⟩) × 1024 and pick the closest option.
  intermediate · programming · The Diffuser (Inversion About Mean)
• Numericcorenew
  For this circuit, what is the probability that measuring all 2 qubits yields exactly |01⟩ (qubit 0 leftmost)?
  intermediate · programming · The Diffuser (Inversion About Mean)
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |10⟩?
  intermediate · programming · One Grover Iteration
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|11⟩)?
  intermediate · programming · Two-Qubit Grover
• Multiple choicecorenew
  Grover search over $N = 16$ items with $M = 1$ solution. The optimal iteration count is $\lfloor (\pi/4)\sqrt{N/M} \rfloor$. How many iterations?
  intermediate · programming · Optimal Number of Iterations
• Numericcorenew
  After running this circuit, what is the probability of measuring |00⟩?
  intermediate · programming · Over-Rotation
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 6073), how many shots land on |00⟩? Estimate P(|00⟩) × 1024 and pick the closest option.
  intermediate · programming · Over-Rotation
• Numericcorenew
  For this circuit, what is the probability that measuring all 2 qubits yields exactly |01⟩ (qubit 0 leftmost)?
  intermediate · programming · Over-Rotation
• Numericcorenew
  After running this circuit, what is the probability of measuring |000⟩?
  intermediate · programming · Multiple Marked Items
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 8466), how many shots land on |000⟩? Estimate P(|000⟩) × 1024 and pick the closest option.
  intermediate · programming · Multiple Marked Items
• Numericcorenew
  For this circuit, what is the probability that measuring all 3 qubits yields exactly |111⟩ (qubit 0 leftmost)?
  intermediate · programming · Multiple Marked Items
• Multiple choicecorenew
  Per this lesson, Grover's quadratic speedup is optimal: what is the proven lower bound on the number of oracle queries any quantum algorithm needs for unstructured search of N items?
  intermediate · programming · Quadratic Speedup
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|101⟩)?
  intermediate · programming · Three-Qubit Grover
• Predictcorenew
  After running this circuit, which outcome has the highest probability?
  intermediate · programming · Three-Qubit Grover
• Numericcorenew
  After running this circuit, what is the expectation value ⟨Z⟩ of qubit 0 (its Bloch Z-component)?
  intermediate · programming · Three-Qubit Grover
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 4559), how many shots land on |101⟩? Estimate P(|101⟩) × 1024 and pick the closest option.
  intermediate · programming · Three-Qubit Grover
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |01⟩?
  intermediate · programming · Checkpoint: Build Grover Search
• Multiple choicecorenew
  Per this lesson, Grover search is the special case of amplitude amplification in which the initial algorithm A and success probability p take what values?
  intermediate · programming · Generalizing Grover
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|00⟩)?
  intermediate · programming · The Amplification Operator
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 8061). Predict how many give |00⟩. Estimate P(|00⟩) × 1024 and pick the closest option.
  intermediate · programming · The Amplification Operator
• Numericcorenew
  After running this circuit, what is the probability that all 2 qubits together read |01⟩ (qubit 0 leftmost)?
  intermediate · programming · The Amplification Operator
• Multiple choicecorenew
  In Grover's algorithm the oracle reflects the state about $|\psi_{bad}\rangle$ and the diffuser reflects it about $|s\rangle$, where the angle between these two axes is $\theta$. Using the geometry fact that composing two reflections whose axes meet at angle $\phi$ gives a rotation by $2\phi$, by what angle does one full Grover iteration (oracle then diffuser) rotate the state?
  intermediate · programming · Reflections as Rotations
• Numericcorenew
  After running this circuit, what is the probability of measuring |10⟩?
  intermediate · programming · Boosting a Known Subroutine
• Multiple choicecorenew
  Per this lesson, what core problem does fixed-point amplification solve compared with standard Grover-style amplitude amplification?
  intermediate · programming · Fixed-Point Amplification
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|011⟩)?
  intermediate · programming · Amplitude Estimation (Intro)
• Multiple choicecorenew
  Quantum counting on a database of $N = 4$ items measures a rotation half-angle $\theta = \pi/6$. The estimated number of solutions is $M = N\sin^2(\theta)$. What is $M$?
  intermediate · programming · Counting Solutions
• Multiple choicecorenew
  Per this lesson, given a circuit A whose good outputs occur with probability p, how many applications of A does amplitude amplification need to find a good output, versus classical repetition?
  intermediate · programming · Applications in Speedups
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|01⟩)?
  intermediate · programming · Combining with Oracles
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |10⟩?
  intermediate · programming · Checkpoint: Amplify an Amplitude
• Multiple choicecorenew
  Shor's algorithm does not factor N directly. Instead it reduces factoring to which problem, solved on the quantum computer, where given a with gcd(a, N) = 1 one seeks the smallest positive r with a^r = 1 (mod N)?
  intermediate · programming · Factoring and Cryptography
• Multiple choicecorenew
  Once Shor's algorithm has found an even order r of a modulo N (with a^(r/2) not congruent to -1 mod N), the identity a^r - 1 = (a^(r/2) - 1)(a^(r/2) + 1) = 0 (mod N) is used. How is a non-trivial factor of N then extracted?
  intermediate · programming · Reduction to Period Finding
• Multiple choicecorenew
  In Shor's order-finding, compute the modular exponential 7^4 mod 15.
  intermediate · programming · Modular Exponentiation
• Multiple choicecorenew
  In Shor's period-finding subroutine, after applying the modular-exponentiation oracle U_f to get sum_x |x>|a^x mod N>, measuring the OUTPUT register collapses the input register to which kind of state?
  intermediate · programming · The Period-Finding Subroutine
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |000⟩?
  intermediate · programming · Using the QFT to Find Periods
• Numericcorenew
  After running this circuit, what is the expectation value ⟨X⟩ of qubit 0 (its Bloch X-component)?
  intermediate · programming · Using the QFT to Find Periods
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 5040). Predict how many give |000⟩. Estimate P(|000⟩) × 1024 and pick the closest option.
  intermediate · programming · Using the QFT to Find Periods
• Numericcorenew
  For this circuit, what is the probability that measuring all 3 qubits yields exactly |010⟩ (qubit 0 leftmost)?
  intermediate · programming · Using the QFT to Find Periods
• Multiple choicecorenew
  A phase-estimation register of size Q = 256 yields the measured value m = 63. Using the continued-fraction expansion of m/Q = 63/256 with denominators bounded by N = 15, what is the recovered period (best denominator ≤ 15)?
  intermediate · programming · Continued Fractions
• Multiple choicecorenew
  Shor's algorithm factors N = 21 with base a = 2 and period r = 6. A factor candidate is gcd(a^(r/2) - 1, N) = gcd(2^3 - 1, 21). What is it?
  intermediate · programming · From Period to Factors
• Multiple choicecorenew
  Shor's algorithm factors $N = 15$ using base $a = 7$ with found period $r = 4$. A factor is $\gcd(a^{r/2} - 1, N) = \gcd(7^2 - 1, 15)$. What is this factor?
  intermediate · programming · Factoring 15
• Multiple choicecorenew
  In Shor's period-finding circuit for factoring an n-bit number N, why does the input register use 2n qubits rather than just n?
  intermediate · programming · Resource Requirements
• Multiple choicecorenew
  How do the asymptotic running times of Shor's algorithm and the classical General Number Field Sieve (GNFS) compare when factoring an n-bit number N?
  intermediate · programming · Why It Beats Classical Factoring
• True / Falsecorenew
  True or false: 'Post-quantum cryptography' means the encryption algorithm itself must run on a quantum computer.
  intermediate · programming · Post-Quantum Cryptography
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |000⟩?
  intermediate · programming · Checkpoint: Period Finding
• Numericcorenew
  After running this circuit, what is the expectation value ⟨X⟩ of qubit 0 (its Bloch X-component)?
  intermediate · programming · Checkpoint: Period Finding
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 8842). Predict how many give |000⟩. Estimate P(|000⟩) × 1024 and pick the closest option.
  intermediate · programming · Checkpoint: Period Finding
• Numericcorenew
  For this circuit, what is the probability that measuring all 3 qubits yields exactly |100⟩ (qubit 0 leftmost)?
  intermediate · programming · Checkpoint: Period Finding
• Multiple choicecorenew
  A binary operation makes a set G a group only if it satisfies four axioms. Which set of four properties are the group axioms?
  intermediate · programming · Group Theory Essentials
• Multiple choicecorenew
  The Hidden Subgroup Problem gives a black-box function $f : G \to S$. What property must $f$ have, relative to the hidden subgroup $H \leq G$, for the problem to be well-posed?
  intermediate · programming · The Hidden Subgroup Problem
• Multiple choicecorenew
  In the Deutsch-Jozsa algorithm cast as an HSP over $G = \mathbb{Z}_2^n$, the final $H^{\otimes n}$ gives the all-zeros outcome amplitude $\alpha = (1/2^n) \sum_x (-1)^{f(x)}$. What does measuring $0^n$ tell you?
  intermediate · programming · Deutsch–Jozsa as HSP
• Multiple choicecorenew
  In Simon's problem, the oracle $f : \{0,1\}^n \to \{0,1\}^n$ satisfies the promise $f(x) = f(y)$ if and only if $y = x \oplus s$ for a secret string $s$. When $s$ is NOT the all-zeros string, what is the structure of $f$?
  intermediate · programming · Simon’s Problem
• Numericcorenew
  After running this circuit, what is the probability of measuring |0000⟩?
  intermediate · programming · Simon’s Algorithm
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 3181), how many shots land on |0000⟩? Estimate P(|0000⟩) × 1024 and pick the closest option.
  intermediate · programming · Simon’s Algorithm
• Numericcorenew
  For this circuit, what is the probability that measuring all 4 qubits yields exactly |0010⟩ (qubit 0 leftmost)?
  intermediate · programming · Simon’s Algorithm
• Multiple choicecorenew
  In Simon's algorithm on n = 3 bits, measurements give the strings y = 010 and y = 111. The hidden string s ≠ 000 satisfies y·s = 0 (mod 2, bitwise) for every sample. Which value of s (as an integer) is the unique nonzero solution?
  intermediate · programming · Solving the Linear System
• Multiple choicecorenew
  When Shor's period finding for $f(x) = a^x \bmod N$ is cast as a Hidden Subgroup Problem over the group $G = \mathbb{Z}_M$ (with $M \geq N^2$ a power of two), what is the hidden subgroup $H$ whose generator reveals the order $r$?
  intermediate · programming · Period Finding as HSP
• Multiple choicecorenew
  For the Hidden Subgroup Problem (HSP), which statement correctly contrasts the abelian and non-abelian cases?
  intermediate · programming · Abelian vs Non-Abelian HSP
• Numericcorenew
  After running this circuit, what is the probability of measuring |000⟩?
  intermediate · programming · Checkpoint: Simon’s Algorithm
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 5340), how many shots land on |000⟩? Estimate P(|000⟩) × 1024 and pick the closest option.
  intermediate · programming · Checkpoint: Simon’s Algorithm
• Numericcorenew
  For this circuit, what is the probability that measuring all 3 qubits yields exactly |001⟩ (qubit 0 leftmost)?
  intermediate · programming · Checkpoint: Simon’s Algorithm
• True / Falsecorenew
  True or false: for a classical fair-coin random walk on a line, the position after $t$ steps has variance $\mathrm{Var}(X_t) = t$, so the typical distance from the origin grows as $\sqrt{t}$ (diffusive spreading).
  intermediate · programming · Classical Random Walks
• Multiple choicecorenew
  A classical random walk on a line spreads diffusively with standard deviation $\sigma \sim \sqrt{t}$. For the Hadamard discrete-time quantum walk, how does the position spread with the number of steps $t$?
  intermediate · programming · Discrete-Time Quantum Walks
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|0⟩)?
  intermediate · programming · The Coin Operator
• Numericcorenew
  This circuit prepares qubit 0 in some state. What expectation value ⟨X⟩ does it have along the Bloch X-axis?
  intermediate · programming · The Coin Operator
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 9184). How many of the shots come out |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  intermediate · programming · The Coin Operator
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |110⟩?
  intermediate · programming · The Shift Operator
• Numericcorenew
  After running this circuit, what is the probability of measuring |000⟩?
  intermediate · programming · Walk on a Line
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 959), how many shots land on |000⟩? Estimate P(|000⟩) × 1024 and pick the closest option.
  intermediate · programming · Walk on a Line
• Numericcorenew
  For this circuit, what is the probability that measuring all 3 qubits yields exactly |110⟩ (qubit 0 leftmost)?
  intermediate · programming · Walk on a Line
• Multiple choicecorenew
  On a line, how does the standard deviation $\sigma$ of the walker's position after $t$ steps scale for a classical random walk versus a discrete-time quantum walk?
  intermediate · programming · Ballistic Spreading
• Multiple choicecorenew
  In the continuous-time quantum walk (CTQW) model of Farhi and Gutmann, the walker evolves on a graph G under the Schrodinger equation. What plays the role of the Hamiltonian?
  intermediate · programming · Continuous-Time Quantum Walks
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|11⟩)?
  intermediate · programming · Walks for Search
• Multiple choicecorenew
  Ambainis's quantum walk on a Johnson graph solves the element distinctness problem (are any two of n numbers equal?) with how many queries, compared with the classical Theta(n)?
  intermediate · programming · Applications of Quantum Walks
• Numericcorenew
  After running this circuit, what is the probability of measuring |000⟩?
  intermediate · programming · Checkpoint: Simulate a Quantum Walk
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 2850), how many shots land on |000⟩? Estimate P(|000⟩) × 1024 and pick the closest option.
  intermediate · programming · Checkpoint: Simulate a Quantum Walk
• Numericcorenew
  For this circuit, what is the probability that measuring all 3 qubits yields exactly |110⟩ (qubit 0 leftmost)?
  intermediate · programming · Checkpoint: Simulate a Quantum Walk
• Multiple choicecorenew
  In first-order Trotterization, e^(-i(H1+H2)t) is approximated as e^(-iH1*t) e^(-iH2*t). The operator error is proportional to the commutator [H1, H2]. What happens when H1 and H2 commute?
  intermediate · programming · Simulating Quantum Systems
• Multiple choicecorenew
  The lesson derives the bridge between Hamiltonian evolution and gate-level programming for a single Pauli Z. Which identity relates $e^{-i\theta Z}$ to the Rz rotation gate?
  intermediate · programming · Time Evolution Operators
• Multiple choicecorenew
  A Hamiltonian is written as a weighted sum of Pauli strings: $H = 0.5\,ZZ - 1.2\,XI + 0.3\,IY$. How many Pauli-string terms does this Hamiltonian contain?
  intermediate · programming · Pauli Hamiltonians
• Multiple choicecorenew
  To implement the two-qubit evolution $e^{-i\theta ZZ}$, the lesson reduces it to a single Rz using a CNOT ladder. Which gate sequence (qubit 0 is control) implements $e^{-i\theta ZZ}$?
  intermediate · programming · Evolving Under a Single Pauli
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|00⟩)?
  intermediate · programming · The Trotter Product Formula
• Numericcorenew
  This circuit prepares qubit 0 in some state. What expectation value ⟨Y⟩ does it have along the Bloch Y-axis?
  intermediate · programming · The Trotter Product Formula
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 620). How many of the shots come out |00⟩? Estimate P(|00⟩) × 1024 and pick the closest option.
  intermediate · programming · The Trotter Product Formula
• Numericcorenew
  Run this circuit, then measure all 2 qubits at once. What is the probability of the joint outcome |10⟩ (qubit 0 leftmost)?
  intermediate · programming · The Trotter Product Formula
• Multiple choicecorenew
  A first-order Trotter step for time $t = 1.0$ has leading error bounded by $(t^2/2)\,\lVert[A,B]\rVert$, where the commutator norm $\lVert[A,B]\rVert = 2$. What is this error bound?
  intermediate · programming · Trotter Error
• Multiple choicecorenew
  For H = A + B, the symmetric (second-order) Suzuki-Trotter formula is S2(tau) = e^(-iA*tau/2) e^(-iB*tau) e^(-iA*tau/2). Compared with first-order Trotter, what global error does it achieve after r steps for total time t?
  intermediate · programming · Higher-Order Trotter
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|00⟩)?
  intermediate · programming · Simulating the Transverse-Field Ising Model
• Predictcorenew
  After running this circuit, which outcome has the highest probability?
  intermediate · programming · Simulating the Transverse-Field Ising Model
• Numericcorenew
  After running this circuit, what is the expectation value ⟨Y⟩ of qubit 0 (its Bloch Y-component)?
  intermediate · programming · Simulating the Transverse-Field Ising Model
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 8312), how many shots land on |00⟩? Estimate P(|00⟩) × 1024 and pick the closest option.
  intermediate · programming · Simulating the Transverse-Field Ising Model
• Numericcorenew
  For this circuit, what is the probability that measuring all 2 qubits yields exactly |01⟩ (qubit 0 leftmost)?
  intermediate · programming · Simulating the Transverse-Field Ising Model
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |0⟩?
  intermediate · programming · Measuring Observables After Evolution
• Numericcorenew
  For qubit 0 after this circuit, what is ⟨Y⟩, the Y-axis projection of its Bloch vector?
  intermediate · programming · Measuring Observables After Evolution
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 1372), how many shots land on |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  intermediate · programming · Measuring Observables After Evolution
• Multiple choicecorenew
  For first-order Trotterization simulating to total time t with target error epsilon, how must the number of Trotter steps r scale?
  intermediate · programming · Resource Scaling
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |00⟩?
  intermediate · programming · Checkpoint: Trotterize a Hamiltonian
• Numericcorenew
  After running this circuit, what is the expectation value ⟨Y⟩ of qubit 0 (its Bloch Y-component)?
  intermediate · programming · Checkpoint: Trotterize a Hamiltonian
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 3120). Predict how many give |00⟩. Estimate P(|00⟩) × 1024 and pick the closest option.
  intermediate · programming · Checkpoint: Trotterize a Hamiltonian
• Numericcorenew
  For this circuit, what is the probability that measuring all 2 qubits yields exactly |10⟩ (qubit 0 leftmost)?
  intermediate · programming · Checkpoint: Trotterize a Hamiltonian
• True / Falsecorenew
  In VQE, the variational principle guarantees that for any trial state $|\psi\rangle$ the energy expectation $\langle\psi|H|\psi\rangle$ is greater than or equal to the true ground-state energy $E_0$, so VQE can never report an energy below $E_0$.
  intermediate · programming · Hybrid Quantum-Classical Computing
• Multiple choicecorenew
  A Hamiltonian has eigenvalues $\{-1, 0, 2\}$. By the variational principle, the expectation $\langle\psi|H|\psi\rangle$ for any normalized trial state $|\psi\rangle$ is bounded below by the ground-state energy. What is that lower bound?
  intermediate · programming · The Variational Principle
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|00⟩)?
  intermediate · programming · Ansatz Circuits
• Predictcorenew
  After running this circuit, which outcome has the highest probability?
  intermediate · programming · Ansatz Circuits
• Numericcorenew
  After running this circuit, what is the expectation value ⟨X⟩ of qubit 0 (its Bloch X-component)?
  intermediate · programming · Ansatz Circuits
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 876), how many shots land on |00⟩? Estimate P(|00⟩) × 1024 and pick the closest option.
  intermediate · programming · Ansatz Circuits
• Numericcorenew
  For this circuit, what is the probability that measuring all 2 qubits yields exactly |11⟩ (qubit 0 leftmost)?
  intermediate · programming · Ansatz Circuits
• Multiple choicecorenew
  Two qubits are prepared in the state $|+\rangle \otimes |0\rangle$ (qubit 0 is $|+\rangle$, qubit 1 is $|0\rangle$). What is the expectation value $\langle Z\rangle$ measured on qubit 1?
  intermediate · programming · Measuring Energy Expectation
• Multiple choicecorenew
  A VQE parameter starts at $\theta = 0.4$. A parameter-shift gradient gives energies $E(+) = 0.6$ and $E(-) = 0.2$, so $\mathrm{grad} = (E(+) - E(-))/2$. With learning rate $0.1$, the gradient-descent update $\theta_{\text{new}} = \theta - 0.1\,\mathrm{grad}$ equals what?
  intermediate · programming · Parameter Optimization
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |00⟩?
  intermediate · programming · VQE for H₂ (Toy Model)
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 5794). How many of the shots come out |00⟩? Estimate P(|00⟩) × 1024 and pick the closest option.
  intermediate · programming · VQE for H₂ (Toy Model)
• Numericcorenew
  After this circuit, what is P(|11⟩) for the whole 2-qubit register measured together (qubit 0 leftmost)?
  intermediate · programming · VQE for H₂ (Toy Model)
• Multiple choicecorenew
  For a hardware-efficient ansatz (HEA) with n qubits and L repeated layers of native single- and two-qubit gates, how does the number of variational parameters scale?
  intermediate · programming · Hardware-Efficient Ansätze
• Multiple choicecorenew
  According to McClean et al. (2018), in a barren plateau the gradient of the VQE cost with respect to a parameter has zero mean and a variance that shrinks how, as a function of the number of qubits n?
  intermediate · programming · Barren Plateaus (Preview)
• Multiple choicecorenew
  A Hamiltonian has the Pauli terms ZI, IZ, ZZ, and XX. Terms that share a common measurement basis can be measured together. How many distinct measurement circuits are needed (Z-basis terms grouped together, XX separately)?
  intermediate · programming · Measuring Many Pauli Terms
• Multiple choicecorenew
  On a real device, the VQE energy $E(\theta)$ is estimated from a finite number of measurement shots. How does the statistical uncertainty $\sigma_E$ in the energy estimate scale with the number of shots $N$?
  intermediate · programming · Convergence and Initialization
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|00⟩)?
  intermediate · programming · Checkpoint: Run a VQE
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 1013). Predict how many give |00⟩. Estimate P(|00⟩) × 1024 and pick the closest option.
  intermediate · programming · Checkpoint: Run a VQE
• Numericcorenew
  After running this circuit, what is the probability that all 2 qubits together read |11⟩ (qubit 0 leftmost)?
  intermediate · programming · Checkpoint: Run a VQE
• Multiple choicecorenew
  In the Max-Cut problem, vertices are partitioned into two sets S and its complement. When is an edge (u, v) counted as 'cut'?
  intermediate · programming · Combinatorial Optimization Problems
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |00⟩?
  intermediate · programming · Cost Hamiltonians
• Numericcorenew
  For qubit 0 after this circuit, what is ⟨X⟩, the X-axis projection of its Bloch vector?
  intermediate · programming · Cost Hamiltonians
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 3730), how many shots land on |00⟩? Estimate P(|00⟩) × 1024 and pick the closest option.
  intermediate · programming · Cost Hamiltonians
• Numericcorenew
  After running this circuit, what is the probability that all 2 qubits together read |01⟩ (qubit 0 leftmost)?
  intermediate · programming · Cost Hamiltonians
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|00⟩)?
  intermediate · programming · The Mixing Hamiltonian
• Numericcorenew
  This circuit prepares qubit 0 in some state. What expectation value ⟨Y⟩ does it have along the Bloch Y-axis?
  intermediate · programming · The Mixing Hamiltonian
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 697). How many of the shots come out |00⟩? Estimate P(|00⟩) × 1024 and pick the closest option.
  intermediate · programming · The Mixing Hamiltonian
• Numericcorenew
  Run this circuit, then measure all 2 qubits at once. What is the probability of the joint outcome |01⟩ (qubit 0 leftmost)?
  intermediate · programming · The Mixing Hamiltonian
• Numericcorenew
  After running this circuit, what is the probability of measuring |00⟩?
  intermediate · programming · The QAOA Circuit
• Numericcorenew
  Run this circuit. What is qubit 0's Bloch X-component — the expectation value ⟨X⟩?
  intermediate · programming · The QAOA Circuit
• Predictcorenew
  On the seeded simulator (seed 2323), this circuit runs for 1024 shots. About how many come out |00⟩? Estimate P(|00⟩) × 1024 and pick the closest option.
  intermediate · programming · The QAOA Circuit
• Numericcorenew
  After this circuit, what is P(|01⟩) for the whole 2-qubit register measured together (qubit 0 leftmost)?
  intermediate · programming · The QAOA Circuit
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |00⟩?
  intermediate · programming · Parameterizing Layers
• Numericcorenew
  After running this circuit, what is the expectation value ⟨X⟩ of qubit 0 (its Bloch X-component)?
  intermediate · programming · Parameterizing Layers
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 5356). Predict how many give |00⟩. Estimate P(|00⟩) × 1024 and pick the closest option.
  intermediate · programming · Parameterizing Layers
• Numericcorenew
  For this circuit, what is the probability that measuring all 2 qubits yields exactly |11⟩ (qubit 0 leftmost)?
  intermediate · programming · Parameterizing Layers
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|011⟩)?
  intermediate · programming · Max-Cut with QAOA
• Numericcorenew
  This circuit prepares qubit 0 in some state. What expectation value ⟨X⟩ does it have along the Bloch X-axis?
  intermediate · programming · Max-Cut with QAOA
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 8389). How many of the shots come out |011⟩? Estimate P(|011⟩) × 1024 and pick the closest option.
  intermediate · programming · Max-Cut with QAOA
• Numericcorenew
  Run this circuit, then measure all 3 qubits at once. What is the probability of the joint outcome |100⟩ (qubit 0 leftmost)?
  intermediate · programming · Max-Cut with QAOA
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |01⟩?
  intermediate · programming · Measuring and Decoding Solutions
• Predictcorenew
  On the seeded simulator (seed 1449), this circuit runs for 1024 shots. About how many come out |01⟩? Estimate P(|01⟩) × 1024 and pick the closest option.
  intermediate · programming · Measuring and Decoding Solutions
• Numericcorenew
  Run this circuit, then measure all 2 qubits at once. What is the probability of the joint outcome |10⟩ (qubit 0 leftmost)?
  intermediate · programming · Measuring and Decoding Solutions
• Multiple choicecorenew
  In a one-layer QAOA the expected cost is $\langle C\rangle(\gamma, \beta) = 0.5 + 0.25 \sin(2\gamma) \sin(2\beta)$. Optimizing the angles classically, what is the maximum value of $\langle C\rangle$?
  intermediate · programming · Classical Optimization of Angles
• Multiple choicecorenew
  For depth p = 1 on unweighted 3-regular graphs, Farhi et al. derived the optimized Max-Cut approximation ratio achievable by QAOA. What bound does the lesson state?
  intermediate · programming · QAOA vs Classical Heuristics
• Multiple choicecorenew
  A QAOA run on a Max-Cut instance achieves an expected cut value of 3.75, while the true optimum is 5. The approximation ratio is achieved/optimal. What is it?
  intermediate · programming · Approximation Ratios
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |010⟩?
  intermediate · programming · Checkpoint: Solve Max-Cut
• Predictcorenew
  On the seeded simulator (seed 3608), this circuit runs for 1024 shots. About how many come out |010⟩? Estimate P(|010⟩) × 1024 and pick the closest option.
  intermediate · programming · Checkpoint: Solve Max-Cut
• Numericcorenew
  Run this circuit, then measure all 3 qubits at once. What is the probability of the joint outcome |101⟩ (qubit 0 leftmost)?
  intermediate · programming · Checkpoint: Solve Max-Cut
• Multiple choicecorenew
  The lesson lists three ways to encode classical features x into qubits. Which description matches AMPLITUDE encoding?
  intermediate · programming · What Is Quantum Machine Learning?
• Numericcorenew
  After running this circuit, what is the probability of measuring |00⟩?
  intermediate · programming · Encoding Classical Data
• Predictcorenew
  Run this circuit. Which basis-state outcome appears most often?
  intermediate · programming · Encoding Classical Data
• Numericcorenew
  For qubit 0 after this circuit, what is ⟨X⟩, the X-axis projection of its Bloch vector?
  intermediate · programming · Encoding Classical Data
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 6451). Predict how many give |00⟩. Estimate P(|00⟩) × 1024 and pick the closest option.
  intermediate · programming · Encoding Classical Data
• Numericcorenew
  After running this circuit, what is the probability that all 2 qubits together read |10⟩ (qubit 0 leftmost)?
  intermediate · programming · Encoding Classical Data
• Numericcorenew
  After running this circuit, what is the probability of measuring |00⟩?
  intermediate · programming · Angle and Amplitude Encoding
• Predictcorenew
  Run this circuit. Which basis-state outcome appears most often?
  intermediate · programming · Angle and Amplitude Encoding
• Numericcorenew
  For qubit 0 after this circuit, what is ⟨X⟩, the X-axis projection of its Bloch vector?
  intermediate · programming · Angle and Amplitude Encoding
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 6600). Predict how many give |00⟩. Estimate P(|00⟩) × 1024 and pick the closest option.
  intermediate · programming · Angle and Amplitude Encoding
• Numericcorenew
  After running this circuit, what is the probability that all 2 qubits together read |10⟩ (qubit 0 leftmost)?
  intermediate · programming · Angle and Amplitude Encoding
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |00⟩?
  intermediate · programming · Parameterized Quantum Circuits
• Predictcorenew
  Which measurement outcome is most likely for this circuit?
  intermediate · programming · Parameterized Quantum Circuits
• Numericcorenew
  This circuit prepares qubit 0 in some state. What expectation value ⟨X⟩ does it have along the Bloch X-axis?
  intermediate · programming · Parameterized Quantum Circuits
• Predictcorenew
  On the seeded simulator (seed 9633), this circuit runs for 1024 shots. About how many come out |00⟩? Estimate P(|00⟩) × 1024 and pick the closest option.
  intermediate · programming · Parameterized Quantum Circuits
• Numericcorenew
  Run this circuit, then measure all 2 qubits at once. What is the probability of the joint outcome |11⟩ (qubit 0 leftmost)?
  intermediate · programming · Parameterized Quantum Circuits
• Multiple choicecorenew
  The lesson notes that plain angle encoding, $|\phi(x)\rangle =$ tensor of $R_y(x_j)|0\rangle$, induces the kernel $K(x,x') =$ product of $\cos^2((x_j - x'_j)/2)$. Why does angle encoding alone give no quantum advantage?
  intermediate · programming · Quantum Feature Maps
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |0⟩?
  intermediate · programming · Quantum Kernels
• Numericcorenew
  For qubit 0 after this circuit, what is ⟨X⟩, the X-axis projection of its Bloch vector?
  intermediate · programming · Quantum Kernels
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 5726), how many shots land on |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  intermediate · programming · Quantum Kernels
• Numericcorenew
  After running this circuit, what is the probability of measuring |00⟩?
  intermediate · programming · Training a Quantum Classifier
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 8759), how many shots land on |00⟩? Estimate P(|00⟩) × 1024 and pick the closest option.
  intermediate · programming · Training a Quantum Classifier
• Numericcorenew
  For this circuit, what is the probability that measuring all 2 qubits yields exactly |11⟩ (qubit 0 leftmost)?
  intermediate · programming · Training a Quantum Classifier
• Multiple choicecorenew
  A variational cost is $E(\theta) = \cos(\theta)$, so its gradient is $dE/d\theta = -\sin(\theta)$. At $\theta = \pi/3$, what are $(E, dE/d\theta)$?
  intermediate · programming · Cost Functions and Gradients
• Multiple choicecorenew
  The parameter-shift rule estimates a gradient as $0.5 ( f(\theta + \pi/2) - f(\theta - \pi/2) )$, where here $f(\theta) = \cos(\theta)$. At $\theta = \pi/4$, what value does it give?
  intermediate · programming · The Parameter-Shift Rule
• True / Falsecorenew
  Expressibility and trainability in variational quantum circuits pull in opposite directions: a highly expressible (random, deep, all-to-all entangling) circuit tends to suffer barren plateaus, while shallow, local, structured circuits retain larger gradients.
  intermediate · programming · Expressivity and Trainability
• Numericcorenew
  After running this circuit, what is the probability of measuring |11⟩?
  intermediate · programming · Checkpoint: A Tiny Quantum Classifier
• Predictcorenew
  Run this circuit. Which basis-state outcome appears most often?
  intermediate · programming · Checkpoint: A Tiny Quantum Classifier
• Numericcorenew
  For qubit 0 after this circuit, what is ⟨X⟩, the X-axis projection of its Bloch vector?
  intermediate · programming · Checkpoint: A Tiny Quantum Classifier
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 945). Predict how many give |11⟩. Estimate P(|11⟩) × 1024 and pick the closest option.
  intermediate · programming · Checkpoint: A Tiny Quantum Classifier
• Numericcorenew
  After running this circuit, what is the probability that all 2 qubits together read |00⟩ (qubit 0 leftmost)?
  intermediate · programming · Checkpoint: A Tiny Quantum Classifier
• Multiple choicecorenew
  A qubit has relaxation time T1 and dephasing (coherence) time T2, related by 1/T2 = 1/(2*T1) + 1/Tphi, where Tphi is the pure dephasing time. What constraint does this place on T2?
  intermediate · programming · Why Real Qubits Are Noisy
• Multiple choicecorenew
  A qubit is in the mixed (classically random) state that is $|0\rangle$ with probability $0.7$ and $|1\rangle$ with probability $0.3$, so $\rho = \mathrm{diag}(0.7, 0.3)$. The purity is $\mathrm{Tr}(\rho^2) = 0.7^2 + 0.3^2$. What is the purity?
  intermediate · programming · Mixed States and Density Matrices
• Multiple choicecorenew
  A bit-flip channel acts on $|0\rangle$ as $\rho' = (1-p)|0\rangle\langle 0| + p|1\rangle\langle 1|$, applying $X$ with probability $p = 0.3$. What is $P(\text{measuring } |1\rangle)$ on the output?
  intermediate · programming · The Bit-Flip Channel
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|0⟩)?
  intermediate · programming · The Phase-Flip Channel
• Numericcorenew
  This circuit prepares qubit 0 in some state. What expectation value ⟨X⟩ does it have along the Bloch X-axis?
  intermediate · programming · The Phase-Flip Channel
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 8154). How many of the shots come out |0⟩? Estimate P(|0⟩) × 1024 and pick the closest option.
  intermediate · programming · The Phase-Flip Channel
• Multiple choicecorenew
  For a single qubit, a depolarizing channel with parameter p shrinks any Bloch-vector component by the factor (1 - 2p/3). With p = 3/4, what is this shrink factor?
  intermediate · programming · Depolarizing Noise
• Multiple choicecorenew
  Under an amplitude-damping (T1) channel with damping probability gamma = 0.5, the ground-state population transforms as rho00' = rho00 + gamma * rho11. For a maximally mixed qubit (rho00 = rho11 = 0.5), what is rho00'?
  intermediate · programming · Amplitude Damping (T₁)
• Multiple choicecorenew
  A dephasing (T2) channel with flip probability p = 0.15 shrinks the off-diagonal coherence as rho01' = (1 - 2p) * rho01. If rho01 = 0.5 initially, what is rho01' after the channel?
  intermediate · programming · Dephasing (T₂)
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|01⟩)?
  intermediate · programming · Simulating Noisy Circuits
• Numericcorenew
  This circuit prepares qubit 0 in some state. What expectation value ⟨X⟩ does it have along the Bloch X-axis?
  intermediate · programming · Simulating Noisy Circuits
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 340). How many of the shots come out |01⟩? Estimate P(|01⟩) × 1024 and pick the closest option.
  intermediate · programming · Simulating Noisy Circuits
• Numericcorenew
  Run this circuit, then measure all 2 qubits at once. What is the probability of the joint outcome |11⟩ (qubit 0 leftmost)?
  intermediate · programming · Simulating Noisy Circuits
• Multiple choicecorenew
  A qubit prepared in $|+\rangle = (|0\rangle+|1\rangle)/\sqrt{2}$ passes through a dephasing channel with $p = 0.3$, giving $\rho$ with populations $0.5$ and off-diagonals $0.5(1-2p)$. The fidelity with $|+\rangle$ is $F = \langle +|\rho|+\rangle = 0.5(1 + 2 \cdot 0.5(1-2p)) = 1 - p$. What is $F$?
  intermediate · programming · Fidelity of Noisy States
• Multiple choicecorenew
  A qubit starts in $|0\rangle$ and passes through two independent bit-flip channels, each flipping the state with probability $p = 0.3$. It ends in $|0\rangle$ only if neither or both flips fire: $P(0) = (1-p)^2 + p^2$. What is $P(\text{measuring } |0\rangle)$ at the end?
  intermediate · programming · Checkpoint: Model a Noisy Qubit
• Multiple choicecorenew
  A single bit is protected by the 3-bit repetition code ($0 \to 000$, $1 \to 111$) and decoded by majority vote; each bit flips independently with probability $p$. What is the probability of a logical (majority) error?
  intermediate · programming · Classical Error Correction
• Multiple choicecorenew
  The no-cloning proof assumes a unitary $U$ with $U|\psi\rangle|0\rangle = |\psi\rangle|\psi\rangle$ for every state. Taking the inner product of this relation applied to two states $|\psi\rangle$ and $|\phi\rangle$ yields which equation, and what does it imply?
  intermediate · programming · Why Quantum Errors Are Different
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |000⟩?
  intermediate · programming · The Three-Qubit Bit-Flip Code
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 998). How many of the shots come out |000⟩? Estimate P(|000⟩) × 1024 and pick the closest option.
  intermediate · programming · The Three-Qubit Bit-Flip Code
• Numericcorenew
  After this circuit, what is P(|111⟩) for the whole 3-qubit register measured together (qubit 0 leftmost)?
  intermediate · programming · The Three-Qubit Bit-Flip Code
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|101⟩)?
  intermediate · programming · Syndrome Measurement
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |000⟩?
  intermediate · programming · Correcting a Bit Flip
• Numericcorenew
  After running this circuit, what is the probability of measuring |000⟩?
  intermediate · programming · The Three-Qubit Phase-Flip Code
• Predictcorenew
  Sampling this circuit 1024 times on the seeded simulator (seed 124), how many shots land on |000⟩? Estimate P(|000⟩) × 1024 and pick the closest option.
  intermediate · programming · The Three-Qubit Phase-Flip Code
• Numericcorenew
  For this circuit, what is the probability that measuring all 3 qubits yields exactly |011⟩ (qubit 0 leftmost)?
  intermediate · programming · The Three-Qubit Phase-Flip Code
• Multiple choicecorenew
  A stabilizer code with minimum distance d is used for error detection and correction. How many single-qubit errors can it detect, and how many can it correct?
  intermediate · programming · Detecting vs Correcting
• Multiple choicecorenew
  In the three-qubit bit-flip code with logical states |0_L> = |000> and |1_L> = |111>, which physical operation implements the logical X gate (X_L) transversally?
  intermediate · programming · Logical vs Physical Qubits
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |01011⟩?
  intermediate · programming · Checkpoint: Detect an Error
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |11⟩?
  intermediate · programming · Project: Benchmark Grover
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|000⟩)?
  intermediate · programming · Project: Build a QFT Library
• Numericcorenew
  This circuit prepares qubit 0 in some state. What expectation value ⟨X⟩ does it have along the Bloch X-axis?
  intermediate · programming · Project: Build a QFT Library
• Predictcorenew
  This circuit is sampled for 1024 shots on the seeded simulator (seed 136). How many of the shots come out |000⟩? Estimate P(|000⟩) × 1024 and pick the closest option.
  intermediate · programming · Project: Build a QFT Library
• Numericcorenew
  Run this circuit, then measure all 3 qubits at once. What is the probability of the joint outcome |001⟩ (qubit 0 leftmost)?
  intermediate · programming · Project: Build a QFT Library
• Numericcorenew
  After running this circuit, what is the probability of measuring |101⟩?
  intermediate · programming · Project: Estimate a Phase Precisely
• Numericcorenew
  Run this circuit, then measure. With what probability do you get |01⟩?
  intermediate · programming · Project: Mini-VQE
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|01⟩)?
  intermediate · programming · Project: QAOA Tuning
• Predictcorenew
  Run 1024 seeded shots of this circuit (seed 7828). Predict how many give |01⟩. Estimate P(|01⟩) × 1024 and pick the closest option.
  intermediate · programming · Project: QAOA Tuning
• Numericcorenew
  After running this circuit, what is the probability that all 2 qubits together read |10⟩ (qubit 0 leftmost)?
  intermediate · programming · Project: QAOA Tuning
• Numericcorenew
  Execute the circuit and measure. How likely is the outcome |00⟩?
  intermediate · programming · Project: Noisy vs Ideal Comparison
• Predictcorenew
  On the seeded simulator (seed 888), this circuit runs for 1024 shots. About how many come out |00⟩? Estimate P(|00⟩) × 1024 and pick the closest option.
  intermediate · programming · Project: Noisy vs Ideal Comparison
• Numericcorenew
  Run this circuit, then measure all 2 qubits at once. What is the probability of the joint outcome |11⟩ (qubit 0 leftmost)?
  intermediate · programming · Project: Noisy vs Ideal Comparison
• Numericcorenew
  After running this circuit, what is the probability of measuring |000⟩?
  intermediate · programming · Project: A Random-Circuit Sampler
• Numericcorenew
  Run this circuit. What is qubit 0's Bloch X-component — the expectation value ⟨X⟩?
  intermediate · programming · Project: A Random-Circuit Sampler
• Predictcorenew
  On the seeded simulator (seed 3921), this circuit runs for 1024 shots. About how many come out |000⟩? Estimate P(|000⟩) × 1024 and pick the closest option.
  intermediate · programming · Project: A Random-Circuit Sampler
• Numericcorenew
  After this circuit, what is P(|001⟩) for the whole 3-qubit register measured together (qubit 0 leftmost)?
  intermediate · programming · Project: A Random-Circuit Sampler
• Multiple choicecorenew
  Grover's algorithm searches for a marked item among N = 2^n unstructured candidates. How many oracle calls does it need, compared with the classical cost?
  intermediate · programming · Reflection: Algorithmic Toolkit
• Numericcorenew
  This circuit is measured once in the computational basis. What is P(|10⟩)?
  intermediate · programming · Capstone: Design an Algorithm Demo