Conserving Magnetization in the XY Ladder with Protected Strings
Conserving Magnetization in the XY Ladder with Protected Strings¶
The two-leg XY ladder
$$H = J \sum_{\langle i,j\rangle} \left(X_i X_j + Y_i Y_j\right)$$
(nearest-neighbour bonds along both legs and across the rungs) conserves the total magnetization $M = \sum_i Z_i$. In Heisenberg-picture Pauli propagation this is a sum rule on the propagated observable $O(t)$: the single-site magnetizations $a_i(t) = \langle Z_i, O(t)\rangle$ satisfy $\sum_i a_i(t) = \langle M, O(t)\rangle = \text{const}$.
Truncation can break this. By orthonormality of the Pauli basis, the only words
overlapping $M$ are the single-site $Z_i$ themselves, so the conserved charge
leaks precisely when a $Z_i$ at the small-coefficient front is dropped below the
threshold. ppvm's preserve_strings exempts chosen Pauli words from truncation;
protecting the $\{Z_i\}$ pins the magnetization exactly, at the cost of $n$ words.
import matplotlib.pyplot as plt
from ppvm import PauliSum
# two-leg ladder: L rungs, N = 2L qubits; qubit (rung j, leg a) -> j + a*L
L, dt, steps, delta = 10, 0.1, 20, 3e-3
N = 2 * L
site = lambda j, a: j + a * L
bonds = [(site(j, a), site((j + 1) % L, a)) for a in (0, 1) for j in range(L)] # legs (periodic)
bonds += [(site(j, 0), site(j, 1)) for j in range(L)] # rungs
M = PauliSum.new(n_qubits=N, terms=[f"Z{q}" for q in range(N)]) # total magnetization
z_strings = ["I" * q + "Z" + "I" * (N - 1 - q) for q in range(N)] # the single-site Z_i
One first-order Trotter step, applied in reverse bond order because Pauli
propagation runs in the Heisenberg picture (observables are propagated backwards
through the circuit). ppvm's two-qubit rotations use the half-angle convention,
rxx(θ) = exp(-i·θ/2·X_iX_j), so we pass 2 * dt to realise the bond gate
$e^{-i\,dt\,(X_iX_j+Y_iY_j)}$. This gate commutes with $M$, but its rxx factor
alone does not — so we truncate only after the full gate is applied: rxx
runs with truncate=False, and the following ryy performs the truncation. We
seed a unit of magnetization on the central rung and track $\sum_i a_i$ with and
without protecting the $\{Z_i\}$.
def run(preserve):
o = PauliSum.new(
n_qubits=N,
terms=[(f"Z{site(L // 2, 0)}", 0.5), (f"Z{site(L // 2, 1)}", 0.5)], # seed: centre rung
min_abs_coeff=delta,
preserve_strings=z_strings if preserve else None,
)
total_z = [o.overlap(M)]
for _ in range(steps):
for a, b in reversed(bonds): # Heisenberg picture -> reverse circuit order
o.rxx(a, b, theta=2 * dt, truncate=False) # half-angle: 2*dt gives e^{-i dt X_iX_j}, no truncation
o.ryy(a, b, theta=2 * dt) # ... truncate only once the bond gate is complete
total_z.append(o.overlap(M))
return total_z
times = [i * dt for i in range(steps + 1)]
mz_unprotected = run(preserve=False)
mz_protected = run(preserve=True)
print(f"total magnetization at t = {steps * dt:.1f}:")
print(f" preserve = 0 -> {mz_unprotected[-1]:.6f}")
print(f" preserve = 1 -> {mz_protected[-1]:.6f}")
total magnetization at t = 2.0: preserve = 0 -> 0.981968 preserve = 1 -> 1.000000
Without protection the magnetization leaks as the spreading front is truncated; protecting the $\{Z_i\}$ holds it at its initial value to machine precision.
fig, ax = plt.subplots()
ax.plot(times, mz_unprotected, "--", label="preserve = 0")
ax.plot(times, mz_protected, "-", label="preserve = 1")
ax.axhline(1.0, ls=":", c="gray")
ax.set_xlabel("Time $t$")
ax.set_ylabel(r"$\sum_i \langle Z_i, O(t)\rangle$")
ax.set_title("XY ladder — magnetization conservation under truncation")
ax.legend()
plt.tight_layout()
plt.show()
