Checkpoint: Simulate a Spin Chain

This checkpoint ties the module together: you will Trotterize a genuine many-body Hamiltonian, run one evolution step on a three-spin chain, and read out the measurement distribution — combining the product-formula construction, the spin-chain encoding, and the measurement skills from the previous lessons.

The transverse-field Ising model

The transverse-field Ising model (TFIM) is the most-studied solvable many-body system and a standard quantum-simulation benchmark. On an open chain of three spins it is

H = -J\,(Z_0 Z_1 + Z_1 Z_2)\; -\; h\,(X_0 + X_1 + X_2).

The $ZZ$ bonds want neighbouring spins to align (ferromagnetic order); the transverse field $h X$ wants each spin to point along $x$ , i.e. into an equal superposition of up and down. The competition between these non-commuting terms — $[Z_i Z_{i+1}, X_i] \ne 0$ — drives the famous quantum phase transition at $h = J$ , and it is exactly this non-commutativity that forces us to Trotterize.

One Trotter step

Split $H$ into the Ising part and the field part and apply each in turn (first order). Because of the minus signs in $H$ , the exact step carries positive exponents,

e^{-iHt} \approx \underbrace{\prod_{q} e^{\,i\,h t\,X_q}}_{\text{field}}\; \underbrace{\prod_{\langle a,b\rangle} e^{\,i\,J t\,Z_a Z_b}}_{\text{bonds}}.

With the simulator's convention $R_P(\theta) = e^{-i\theta P/2}$ , a positive exponent $e^{+i\phi P}$ is the rotation $R_P(-2\phi)$ , so faithfully realising the minus-sign Hamiltonian needs $\theta_{zz} = -2Jt$ and $\theta_x = -2ht$ .

Each Ising bond $e^{+i J t\, Z_a Z_b} = R_Z(-2Jt)$ in the parity sandwich $\text{CNOT}_{a\to b}\,R_Z(\theta_{zz})_b\,\text{CNOT}_{a\to b}$ , applied to bonds $(0,1)$ and $(1,2)$ .
The transverse field $e^{+i h t\, X_q} = R_X(-2ht)$ on each qubit.

What you will observe

Starting from the polarised state $|000\rangle$ , the Ising bonds alone only add phases and leave the all-zero outcome certain. It is the transverse field that does the interesting work: each $R_X$ rotates a little amplitude into $|1\rangle$ , so after one step the measurement distribution spreads across the basis — most weight stays on $|000\rangle$ , with progressively smaller probabilities on states having one, two, or three flipped spins. The histogram you get is a snapshot of how quantum fluctuations begin to melt the ordered state.

Try it

Build one first-order Trotter step of the 3-spin TFIM ( $J = h = 1$ , $t = 0.3$ ), starting from $|000\rangle$ , then measure. The grader checks the full measurement distribution against the value derived from the reference circuit.

Run your code to see the quantum state.