SU(2) LSH hadron dynamics, 120 qubits: Pauli propagation #250

Name

SU(2) LSH hadron dynamics, 120 qubits: Pauli propagation

Circuit

su2_hadron_dynamics_lsh_x_100_meson

Observable value

60.00129

Error bound (low)

No response

Error bound (high)

No response

Method

Pauli propagation (monoprop)

Method proof

We simulate the two 120-qubit, 20-Trotter-step circuits x_100_meson.qasm and x_100_SCV.qasm exactly as they are given, using Heisenberg-picture Pauli propagation with the monoprop package. The only approximation is a coefficient threshold (atol) applied during propagation.

We use the definitions from submissions #149 and #202. With $n(w) = (1 - <Z_w>)/2$ on output qubit $w$, and site $r$ stored on qubits $2r$ and $2r+1$:

$$ Q=\sum_{w=0}^{119} n(w) = 60-\frac{1}{2}\sum_{w}\langle Z_w\rangle, \qquad n_f=\sum_{r=0}^{59}(-1)^r\Big[\big(n(2r)+n(2r+1)\big)_{\rm meson}-\big(n(2r)+n(2r+1)\big)_{\rm SCV}\Big]. $$

The circuit conserves Q exactly, so Q = 60 for either initial state. The observable value submitted above is Q for the meson circuit, matching the circuit ID. No error bars are provided as there is no sampling involved. There is a systematic error introduced by the coefficient truncation, which can be controlled by decreasing the atol parameter in monoprop's PauliPropagator object.

Method

1. Exact Clifford folding. The circuit body has 21,350 Clifford gates (h, cx, x, and swap for routing) and 10,720 rz gates. qiskit's LitinskiTransformation moves every Clifford to the end of the circuit. Each rz(θ) becomes a single Pauli-product rotation exp(-iθP/2), and one final Clifford D is left over. CommutativeOptimization then merges rotations that repeat the same generator with only commuting rotations between them. That leaves 8,320 rotations. Both steps are exact and take about 0.7 s.
2. Observable in the folded frame. For each output wire we form D† Z_w D, which is a single Pauli string, by reading it off the stabilizer tableau.
3. Pauli propagation. monoprop's PauliPropagator propagates each D† Z_w D backwards through the 8,320 rotations and evaluates it on the product initial state. That is 240 independent propagations (120 qubits × 2 initial states). Terms with |c| < atol are dropped. No Pauli-weight truncation is applied.

Script to reproduce the results (requires qiskit and monoprop): Github gist

Validation at short depth

We truncated the same pipeline after 1, 2 and 3 Trotter steps and ran it at atol = 1e-9. It reproduces the published curve and conserves Q to within 1.3e-7:

Convergence at step 20

These runs were on one Apple M5 (10 cores, 32 GB), with no weight cutoff. The time is the propagation step only. Setup (parsing and folding) adds 0.7 s.

n_f is stable to within 7e-4 from atol = 1e-5 down to 1e-7. We report n_f(20) = 0.1169 ± 0.0007, where the uncertainty is the spread across truncation levels.

Comparison with the other entries for this instance

† #227 reports the differential combination 60 + Σ⟨Z⟩_meson − Σ⟨Z⟩_SCV, not the single-circuit Q above, so its |Q - 60| isn't directly comparable. The same combination from our runs is 60.00034 at atol = 3e-7 and 59.99999 at 1e-7. We submit the meson-only Q because it is the stricter test: truncation errors in the two circuits are strongly correlated, since they share every gate, and so they largely cancel in the difference.
*See comment below by @Roland-Farrell.

Quantum runtime (seconds)

No response

Classical runtime (seconds)

306

Compute resources (quantum)

No response

Compute resources (classical)

Apple M5 CPU (10 cores), 32 GB RAM (Apple Macbook Pro)

Notes

ClaudeCode (Claude Opus 5) helped with this submission

Authors

Matteo A. C. Rossi

Institutions

Algorithmiq