The States |0⟩ and |1⟩

A quantum computer stores information in qubits. Before exploring superposition, you need to know the two states a qubit can be in when you measure it — the computational basis states $|0\rangle$ and $|1\rangle$.

The two basis states

In Dirac (bra-ket) notation these states are written $|0\rangle$ and $|1\rangle$. Behind the notation each state is a two-dimensional complex column vector:

Each entry is a probability amplitude: the probability of a particular measurement outcome equals the squared modulus of the corresponding amplitude. The top entry corresponds to outcome $0$ and the bottom entry to outcome $1$. For $|0\rangle$ the top amplitude is $1$ and the bottom is $0$, giving measurement probabilities $|1|^2 = 1$ and $|0|^2 = 0$ — so a measurement is certain to return 0. For $|1\rangle$ it is the other way around — measurement always returns 1.

These two vectors form an orthonormal basis for the two-dimensional Hilbert space $\mathbb{C}^2$: they are orthogonal ($\langle 0 | 1 \rangle = 0$) and each has unit length ($\langle 0 | 0 \rangle = \langle 1 | 1 \rangle = 1$).

Why column vectors?

Representing states as vectors lets us describe quantum gates as matrix multiplication. A gate $U$ maps a state $|\psi\rangle$ to a new state $U|\psi\rangle$ by ordinary matrix–vector multiplication. Every gate in this course is an operation on the same $\mathbb{C}^2$ (or a tensor product of copies of it), so getting the column-vector picture in your head now pays off everywhere else.

Try it

Qubits always start in $|0\rangle$. Prepare the state $|1\rangle$ by flipping the qubit, then press Check to confirm the simulator agrees with the column vector $\begin{pmatrix}0\\1\end{pmatrix}$.

After running, look at the State Vector panel. The amplitude of $|0\rangle$ is $0$ and the amplitude of $|1\rangle$ is $1$, exactly matching the column vector above.

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