Simulating the Transverse-Field Ising Model

The Transverse-Field Ising Model (TFIM) is one of the simplest many-body Hamiltonians that exhibits a quantum phase transition. It is a canonical test case for Hamiltonian simulation because its two terms — a nearest-neighbour $ZZ$ interaction and a single-qubit $X$ field — do not commute, so Trotterisation introduces a non-trivial approximation error that can be measured and reduced.

The Hamiltonian

For $n$ qubits with open boundary conditions (couplings only between neighbours $i$ and $i+1$ ), the TFIM Hamiltonian is

H = -J \sum_{i=0}^{n-2} Z_i Z_{i+1} - h \sum_{i=0}^{n-1} X_i,

where $J$ is the Ising coupling strength and $h$ is the transverse-field amplitude. For $n=2$ there is exactly one ZZ coupling (between qubits 0 and 1) and two single-qubit $X$ terms:

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

When $J \gg h$ the ground state is a ferromagnetic phase with spins aligned along $Z$ ; when $h \gg J$ the ground state is a paramagnetic phase with spins aligned along $X$ . The quantum phase transition occurs at the critical point $J = h$ .

Trotter decomposition

Because $[ZZ, X \otimes I] \neq 0$ , we cannot exponentiate $H$ exactly in closed form with simple two-local gates. Instead we split the Hamiltonian as $H = H_{ZZ} + H_X$ and apply the first-order Trotter formula over a step $dt$ :

e^{-iH\,dt} \approx e^{-iH_{ZZ}\,dt}\,e^{-iH_X\,dt} + O(dt^2).

Each factor is itself a product of commuting terms, so it factorises exactly.

The ZZ factor

$H_{ZZ} = -J\,Z_0 Z_1$ . The Trotter factor is

e^{-i(-J)\,dt\,Z_0 Z_1} = e^{+iJ\,dt\,Z_0 Z_1}.

Writing this as $e^{-i\theta\,ZZ}$ with $\theta = -J\,dt$ , and recalling the standard decomposition from the previous lesson,

e^{-i\theta\,ZZ} = \mathrm{CNOT}_{0\to1}\,\bigl(I \otimes R_z(2\theta)\bigr)\,\mathrm{CNOT}_{0\to1},

the three-gate circuit is cx(0,1), rz(2*theta, 1), cx(0,1) with $\theta = -J\,dt$ .

The X factor

$H_X = -h(X_0 + X_1)$ . The two terms commute ( $X_0$ and $X_1$ act on different qubits), so

e^{-i(-h)\,dt\,(X_0 + X_1)} = e^{+ih\,dt\,X_0}\,e^{+ih\,dt\,X_1}.

Each factor is $e^{-i\theta X_q}$ with $\theta = -h\,dt$ , which equals $R_x(2\theta) = R_x(-2h\,dt)$ on qubit $q$ .

Circuit for $J=h=1$ , $dt = \pi/8$

With $J = h = 1$ and $dt = \pi/8$ , the angles simplify to $\theta_{ZZ} = -J\,dt = -\pi/8$ and $\theta_X = -h\,dt = -\pi/8$ . The corresponding gate angles are

2\theta_{ZZ} = -\frac{\pi}{4}, \qquad 2\theta_X = -\frac{\pi}{4}.

The complete first-order Trotter step is thus the five-gate sequence

\underbrace{\mathrm{CNOT}_{01}\;R_z\!\bigl(-\tfrac{\pi}{4}\bigr)_1\;\mathrm{CNOT}_{01}}_{e^{+iJ\,dt\,ZZ}} \;\underbrace{R_x\!\bigl(-\tfrac{\pi}{4}\bigr)_0\;R_x\!\bigl(-\tfrac{\pi}{4}\bigr)_1}_{e^{+ih\,dt\,(X_0+X_1)}}.

Try it

Implement one first-order Trotter step for the 2-qubit TFIM with $J = h = 1$ and $dt = \pi/8$ . The grader checks the full unitary matrix, so an empty circuit will not pass. Use the hints if you need a reminder of the gate patterns.

Run your code to see the quantum state.