E — Random SU(4) brickwork

Manual: Truncation

The deliberate worst case. Every other benchmark here is built from a physically structured circuit with commuting layers, Clifford points, or both, so truncation has real structure to exploit. This one draws an independent Haar-random SU(4) block for every brickwork site: no stabilizer structure, no commuting sublattice, no light-cone shortcut past nearest-neighbour causality. Every number below shows how the engine behaves with no help from problem structure.

Setup

n = 36 qubits, brickwork layers alternating on (0,1),(2,3),… and (1,2),(3,4),…, an independent Haar-random SU(4) block per site, observable Z_18 (the central qubit), Heisenberg direction. Seed 20260831, fixed throughout. Single-threaded.

Results

There is no Clifford savings anywhere in this circuit. Two regimes are visible in the numbers:

• Pre-saturation growth. At min_abs_coeff = 1e-4, peak term count goes 15 → 253 → 4038 → 58 079 → 667 088 → 2 296 294 → 4 110 454 for depth 1…7 — roughly an order of magnitude every one to two layers, faster than 4^depth: a generic 2-qubit Haar block’s adjoint action is a dense ~15×15 real orthogonal map on the traceless Pauli algebra, not a 4-way branch.
• Post-saturation collapse. At looser cutoffs the peak plateaus quickly (~1300–1700 terms at 1e-2, ~80 000–110 000 at 1e-3) and the final term count then falls to zero: by depth 10 (1e-2) or depth 20 (1e-3), no coefficient in the operator’s support survives the cutoff. This is the expected fate of a local operator diffusing through a non-integrable, non-Clifford brickwork — ⟨Z_18(t)⟩ reaches the maximally mixed value at truncation resolution.

The 1e-4 grid stops at depth 7 (25.0 s; depth 6 was 7.9 s) — per-step cost grows ~2.3–3× at this cutoff, so depth 8 alone would cost roughly a minute. The two looser truncations already show the full rise–plateau–collapse shape cheaply.

Error against runtime

n = 16, depth 6 (statevector-checkable, deep enough for fanout to be well underway). Oracle: −0.0497243601.

Error falls roughly a decade per decade of cutoff until ~1e-6, where term count and runtime saturate: two more decades of tightness add under 4% more terms and no measurable runtime, but still buy three more decades of accuracy, since the residual error there is dominated by which low-weight terms survive rather than raw term count.

The eps = 1e-2 row is non-monotone: its error (8.1e-2) is larger than the empty-sum row above it (4.97e-2, just |oracle|). Truncation error is not guaranteed monotone in the cutoff for a generic, non-sign-coherent operator.

Time and memory against n

Depth 6, min_abs_coeff = 1e-4, n from 8 to the 36-qubit headline. Oracle-checked up to n = 24; n ∈ {28, 32, 36} is self-converged only (oracle_checked: false in results.json). Wall time tracks term count, not n directly: this circuit’s causal structure means the light cone from the central qubit sets how many terms accumulate by a fixed depth, not n itself.

Every point’s peak_memory_kb_delta reads 0.0 in results.json. RSS is cumulative across a process; the figure is an upper bound when other suites ran first — a real per-point number needs one process per point, as in Benchmark D.

Cross-engine comparison

Schema v1’s unitary_2q gate is exactly the SU(4) block this circuit is built from, and PauliPropagation.jl 0.8.2 defines no Schrödinger transfer map for a TransferMapGate, so only direction="heisenberg" is comparable — the same restriction applies to any unitary_1q/unitary_2q/noise gate against this version.

min_abs_coeff = 1e-4 is deliberately non-dyadic: the one known coefficient-boundary divergence needs exact dyadic coefficients landing on the cutoff, which requires a Clifford point. Every coefficient in a Haar-random SU(4) circuit is an irrational float, so the boundary case cannot arise here.

Per-layer term-count parity holds exactly — matched at both sizes, including the final expectation to 1e-12:

Timed comparison (n = 10, depth 5, single-threaded both sides, warm/JIT-excluded on the jl side):

The two engines are within measurement noise of each other (±5–8% single-threaded is expected between any two runs on this host) — a parity-gated same-ballpark result, not a speed claim in either direction.

Validation

n ∈ {8, 12, 16, 20, 24}, depth 3, no truncation — the light cone at depth 3 has at most 15 non-identity Pauli letters, giving 4095 terms at every size. Agreement is to double-precision tolerance against qiskit Aer’s dense statevector, two engines sharing no simulation code:

Same seed twice gives identical term counts and expectation to 1e-12 (test_benchmark_e_su4.py::test_same_seed_twice_gives_identical_term_counts_and_expectation, always in CI), with a negative control confirming distinct seeds generically disagree.

Numbers: every figure on this page traces to benchmarks/python/su4_staircase/README.md, including the run command, with the raw records in results.json next to it.