Trotterized Time Evolution of the XZZ Ising Chain
Trotterized Time Evolution of the XZZ Ising Chain¶
In this example we simulate the time evolution of the XZZ Ising chain Hamiltonian
$$H = h \sum_i X_i + J \sum_i Z_i Z_{i+1}$$
using first-order Trotterized time evolution and Heisenberg-picture Pauli propagation.
In the Heisenberg picture we propagate observables backwards through the circuit rather than evolving a state vector forwards. ppvm represents the observable as a weighted sum of Pauli strings and applies each gate layer analytically.
One Trotter step of duration $\delta t$ decomposes the unitary as
$$U(\delta t) \approx \prod_i e^{-i h \delta t X_i} \prod_i e^{-i J \delta t Z_i Z_{i+1}}$$
We run two simulations: a noiseless baseline and a noisy version with qubit loss,
using ppvm's LossyPauliSum. Note that the noiseless simulation is still subject
to truncation and therefore an approximate solution.
import matplotlib.pyplot as plt
from ppvm import LossyPauliSum, PauliSum
Parameters¶
n = 6
h = 1.0
j = 1.5 * h
dt = 0.1 / h
time = 3.0 / h
Noiseless simulation¶
We want to compute $\langle \sum_i Z_i \rangle$ on the all-zeros state. In the Heisenberg picture, we initialise the observable as
$$O = \sum_{i=0}^{n-1} Z_i$$
using ppvm's compact notation where "Z3" means $Z$ on qubit 3 and $I$ everywhere else.
Each call to trotter_step applies one Trotter step in reverse (Heisenberg picture):
- RX on every qubit — implements $e^{-i h \delta t X_i}$
- RZZ on every neighbouring pair — implements $e^{-i J \delta t Z_i Z_{i+1}}$
Truncation applies automatically after every gate and channel. There are currently two possible ways to truncate in a loss-less simulation:
- Coefficient truncation: every Pauli string with a leading coefficient, whose absolute value is smaller than
min_abs_coeffis truncated. - Max Pauli Weight truncation: every Pauli string with more than
max_pauli_weightnon-identity Paulis gets truncated.
state = PauliSum.new(
n_qubits=n,
terms=[f"Z{i}" for i in range(n)],
min_abs_coeff=1e-6,
max_pauli_weight=8,
)
theta_x = dt * h
theta_zz = dt * j
def trotter_step(state, n, theta_x, theta_zz):
for i in range(n):
state.rx(i, theta=theta_x)
for i in range(n - 1):
state.rzz(i, i + 1, theta=theta_zz)
Noiseless time evolution¶
steps = int(time / dt)
times = [i * dt for i in range(steps + 1)]
ev_noiseless = []
for _ in range(steps):
ev_noiseless.append(state.overlap_with_zero())
trotter_step(state, n, theta_x, theta_zz)
ev_noiseless.append(state.overlap_with_zero())
print(f"Max Pauli weight (noiseless): {state.current_max_weight()}")
Max Pauli weight (noiseless): 6
Noisy simulation with qubit loss¶
We repeat the simulation using LossyPauliSum, which extends the Pauli basis with a
loss operator $L$ to track qubits that have left the computational subspace.
After each gate layer we apply:
- a single-qubit depolarising channel (
pauli_error) or two-qubit depolarising channel (two_qubit_pauli_error) to model gate errors - a loss channel (
loss_channel) to model qubit loss at the same locations
In addition to the two truncation strategies, we can now also truncate using
max_loss_weight, which removes any Pauli String which has more Ls than
that thresholds. This is justified since these Pauli strings only contribute
little to the final average value.
noise_1q = [
1e-3,
1e-3,
1e-3,
] # symmetric single-qubit depolarising: equal p for X, Y, Z
noise_2q = [
1e-3 / 15.0
] * 15 # symmetric two-qubit depolarising over all 15 non-identity Paulis
p_loss = 1e-3 # loss probability per gate location
noisy_state = LossyPauliSum.new(
n_qubits=n,
terms=[f"Z{i}" for i in range(n)],
min_abs_coeff=1e-6,
max_pauli_weight=8,
max_loss_weight=2,
)
# Reset the loss register on every qubit before propagation: this ensures that qubits
# which become lost during the circuit are counted as |0⟩ when computing overlap_with_zero.
# **NOTE**: without truncation, this scales exponentially
for i in range(n):
noisy_state.reset_loss_channel(i)
def noisy_trotter_step(state, n, theta_x, theta_zz):
for i in range(n):
state.loss_channel(i, p_loss)
state.pauli_error(i, p=noise_1q)
state.rx(i, theta=theta_x)
for i in range(n - 1):
state.loss_channel(i + 1, p_loss)
state.loss_channel(i, p_loss)
state.two_qubit_pauli_error(i, i + 1, p=noise_2q)
state.rzz(i, i + 1, theta=theta_zz)
Noisy time evolution¶
ev_noisy = []
for _ in range(steps):
ev_noisy.append(noisy_state.overlap_with_zero())
noisy_trotter_step(noisy_state, n, theta_x, theta_zz)
ev_noisy.append(noisy_state.overlap_with_zero())
print(f"Max Pauli weight (noisy): {noisy_state.current_max_weight()}")
Max Pauli weight (noisy): 6
Results¶
Both curves start at 1 (all qubits in $|0\rangle$). In the interaction-dominated regime ($J > h$) the magnetisation oscillates before decaying. The noisy simulation decays faster due to depolarisation and qubit loss.
fig, ax = plt.subplots()
ax.plot(times, [ev / n for ev in ev_noiseless], label="noiseless")
ax.plot(times, [ev / n for ev in ev_noisy], label="noisy + loss")
ax.set_xlabel("Time $t$")
ax.set_ylabel(r"$\langle \sum_i Z_i \rangle / n$")
ax.set_title("XZZ Ising chain — Trotterized evolution")
ax.legend()
plt.tight_layout()
plt.show()
