README.md

Stack depth benchmarks

bench_depth.py compares one-shot SRC over a stack (src(A_1, ..., A_k, psi)) with sequential pairwise application (apply from right to left, truncating to chi after every product), as a function of the number of trains in the stack. At depth 2 both methods run the same computation, which makes that row a sanity check.

• accuracy: median relative error against the exact dense product on 10-site chains over 20 random instances, next to lower_bound, the largest best-rank-chi tail over all cuts, which no train of bond dimension chi can beat. one_shot_wins is the fraction of instances where one-shot is more accurate, ignoring ties and instances where both are exact (nan if none is left).
• timing: best-of-3 wall time on 30-site chains, MPS bond 64, chi = 64. ratio is one-shot time over sequential time.
• evolve: quench dynamics of 50+ qubits on the GPU (see Leonardo).

Families: random are complex Gaussian MPOs of bond 3 (flat spectra); trotter-<dt> are brickwork layers of a mixed-field Ising model with random couplings (bond at most 4, decaying spectra). A .mps suffix ends the stack with an MPS. In timing, x<k> . mps is k layers applied to the MPS.

uv run python benches/stack/bench_depth.py accuracy --output accuracy.md
uv run python benches/stack/bench_depth.py timing --output timing.md

Tuple options repeat the flag, e.g. --depths 2 --depths 3.

Leonardo: quench of 50 and 100 qubits

evolve runs the quench of the mixed-field Ising chain H = sum Z_i Z_{i+1} + 1.05 sum X_i + 0.5 sum Z_i from |0...0>, the non-integrable point of Banuls, Cirac and Hastings (PRL 106, 050405, 2011). A first-order Trotter step is a single MPO of bond 2; 80 steps of dt = 0.1 reach t = 8, where the entanglement exceeds what chi = 512 holds. SRC applies k steps per sweep (k = 1 is pairwise application), on one A100-64GB. The reference is k = 1 at chi = 1024; rows give the wall time of the 80 steps, the infidelity of the final state and the error of <Z> on the middle qubit. quimb applies the same MPO with exact SVD truncation on the 32 cores of the node. See HPC clusters for how to run on Leonardo.

uv run python benches/stack/bench_depth.py evolve --n-sites 50 --quimb-max-chi 0
uv run python benches/stack/bench_depth.py evolve --n-sites 100 --quimb-max-chi 0
uv run python benches/stack/bench_depth.py evolve --n-sites 50 --chis 64 --quimb-max-chi 64

qubits	chi	method	seconds	infidelity	z_error
50	64	quimb cpu	731	0.0368	0.00408
50	64	src gpu k=1	6.87	0.0616	0.0134
50	64	src gpu k=2	4.2	0.0621	0.012
50	64	src gpu k=4	2.89	0.0631	0.0135
50	128	src gpu k=1	7.61	0.0212	0.00554
50	128	src gpu k=2	4.49	0.0216	0.006
50	128	src gpu k=4	3.01	0.0218	0.00596
50	256	src gpu k=1	13.9	0.00527	0.00168
50	256	src gpu k=2	7.82	0.00538	0.00191
50	256	src gpu k=4	5.86	0.0055	0.00187
50	512	src gpu k=1	33.9	0.000662	0.000235
50	512	src gpu k=2	19.2	0.000692	0.000259
50	512	src gpu k=4	19.1	0.000711	0.000276
100	64	src gpu k=1	13.1	0.136	0.0126
100	64	src gpu k=2	7.63	0.137	0.0151
100	64	src gpu k=4	5.22	0.138	0.0141
100	128	src gpu k=1	15.7	0.0499	0.00592
100	128	src gpu k=2	8.77	0.0509	0.0064
100	128	src gpu k=4	5.61	0.0512	0.00625
100	256	src gpu k=1	28.2	0.0131	0.00168
100	256	src gpu k=2	15.4	0.0134	0.00166
100	256	src gpu k=4	11.6	0.0137	0.00179
100	512	src gpu k=1	72.3	0.00175	0.000245
100	512	src gpu k=2	40.6	0.00184	0.000254
100	512	src gpu k=4	41.4	0.00188	0.000247

The references take 138 s (50 qubits) and 308 s (100 qubits) and agree on the middle <Z> to 1e-6 (0.667890 and 0.667891).

• Exact SVD truncation (quimb) is more accurate at equal chi: infidelity 0.037 against 0.062 at chi = 64. SRC at chi = 128 is more accurate than quimb at chi = 64, and at chi = 512 reaches 7e-4. The timings compare SRC on one GPU (2.9-6.9 s) with quimb on 32 cores (731 s), and the quimb time looks inflated by how it is called, so they are not a speed-up; a comparison on the same hardware is still to do.
• Fusing two or four Trotter steps per sweep is 1.8-2.8x faster than one step at a time and raises the infidelity by at most 7 %. At chi = 512 four steps are no faster than two: the chi**2 times 2**k cost of the sweep takes over.
• The time grows linearly with the number of qubits; the infidelity of a fixed chi grows with it, while the local error does not.

Small-scale results (laptop)

Measured on an AMD Ryzen 7 7840U, 16 threads, with light background load.

Accuracy

family	depth	chi	lower_bound	one_shot	sequential	one_shot_wins
random.mps	2	4	0.168	0.499	0.499	nan
random.mps	2	8	0.0298	0.0989	0.0989	nan
random.mps	2	16	2.59e-16	1.58e-15	1.58e-15	nan
random.mps	3	4	0.194	0.519	0.636	0.9
random.mps	3	8	0.0584	0.188	0.191	0.75
random.mps	3	16	0.00667	0.0196	0.0194	0.35
random.mps	4	4	0.263	0.611	0.773	0.9
random.mps	4	8	0.0978	0.275	0.335	0.85
random.mps	4	16	0.0145	0.041	0.0533	0.8
random	2	8	0.0734	0.318	0.318	nan
random	2	16	1.3e-15	1.53e-15	1.53e-15	nan
random	2	32	7.34e-16	1.61e-15	1.61e-15	nan
random	3	8	0.289	0.75	0.755	0.7
random	3	16	0.102	0.352	0.351	0.65
random	3	32	9.17e-16	2.19e-15	2.7e-15	nan
random	4	8	0.397	0.828	0.905	1
random	4	16	0.217	0.579	0.607	0.95
random	4	32	0.0819	0.274	0.27	0.55
trotter-0.3.mps	2	2	0.124	0.265	0.265	nan
trotter-0.3.mps	2	4	6.04e-16	1.28e-15	1.28e-15	nan
trotter-0.3.mps	2	8	3.4e-16	1.38e-15	1.38e-15	nan
trotter-0.3.mps	3	2	0.124	0.388	0.432	0.65
trotter-0.3.mps	3	4	0.000264	0.00162	0.00125	0.5
trotter-0.3.mps	3	8	5.53e-16	1.41e-15	1.68e-15	nan
trotter-0.3.mps	4	2	0.15	0.502	0.517	0.45
trotter-0.3.mps	4	4	0.0023	0.0119	0.0103	0.4
trotter-0.3.mps	4	8	2.57e-08	1.16e-07	1.14e-07	0.5
trotter-0.3	2	4	2.87e-15	1.27e-15	1.27e-15	nan
trotter-0.3	2	8	2.2e-15	1.11e-15	1.11e-15	nan
trotter-0.3	2	16	1.38e-15	1.2e-15	1.2e-15	nan
trotter-0.3	3	4	0.000457	0.00324	0.00241	0.5
trotter-0.3	3	8	2.17e-07	2.65e-06	1.76e-06	0.35
trotter-0.3	3	16	1.95e-15	1.47e-15	1.65e-15	nan
trotter-0.3	4	4	0.000522	0.00471	0.00487	0.55
trotter-0.3	4	8	2.45e-07	3.78e-06	4.85e-06	0.6
trotter-0.3	4	16	1.98e-15	1.75e-15	2.4e-15	nan
trotter-0.8.mps	2	4	6.44e-16	1.33e-15	1.33e-15	nan
trotter-0.8.mps	2	8	4.41e-16	1.36e-15	1.36e-15	nan
trotter-0.8.mps	2	16	1.93e-16	1.23e-15	1.23e-15	nan
trotter-0.8.mps	3	4	0.00926	0.0393	0.0393	0.5
trotter-0.8.mps	3	8	6.2e-16	1.45e-15	1.85e-15	nan
trotter-0.8.mps	3	16	2.97e-16	1.53e-15	1.65e-15	nan
trotter-0.8.mps	4	4	0.0647	0.223	0.202	0.4
trotter-0.8.mps	4	8	7.87e-05	0.000315	0.00041	0.55
trotter-0.8.mps	4	16	4.6e-16	1.76e-15	2.27e-15	nan
trotter-0.8	2	8	1.79e-15	1.29e-15	1.29e-15	nan
trotter-0.8	2	16	1.19e-15	1.4e-15	1.4e-15	nan
trotter-0.8	2	32	6.45e-16	1.21e-15	1.21e-15	nan
trotter-0.8	3	8	0.000394	0.00364	0.00357	0.5
trotter-0.8	3	16	1.78e-15	1.59e-15	1.97e-15	nan
trotter-0.8	3	32	8.17e-16	1.7e-15	1.91e-15	nan
trotter-0.8	4	8	0.00048	0.00582	0.00579	0.55
trotter-0.8	4	16	1.73e-15	2.81e-15	4.16e-15	nan
trotter-0.8	4	32	8.41e-16	1.88e-15	2.63e-15	nan

Timing

family	depth	chi	one_shot_s	sequential_s	ratio
trotter-0.3 x1 . mps	2	64	0.0764	0.0642	1.19
trotter-0.3 x2 . mps	3	64	0.0936	0.147	0.639
trotter-0.3 x3 . mps	4	64	0.201	0.207	0.975
trotter-0.3 x4 . mps	5	64	0.384	0.283	1.36
random D=8 x1 . mps	2	64	0.283	0.28	1.01
random D=8 x2 . mps	3	64	1.67	0.52	3.21
random D=8 x3 . mps	4	64	24.7	0.837	29.5
random D=16 x1 . mps	2	64	0.638	0.622	1.03
random D=16 x2 . mps	3	64	11.8	1.31	9.05
random D=16 x3 . mps	4	64	316	2.33	136

Summary

• One sweep over the whole stack is at best moderately more accurate than pairwise application. For random stacks ending in an MPS it has about 20 % lower median error at depth 4 for every chi measured, and at depth 3 only at the smallest chi (at larger chi the gain vanishes); it wins 75-90 % of those instances. For random MPO products the gain is at most 8 % (depth 4, small chi). For Trotter layers there is no systematic difference (wins 35-65 %).
• Both methods sit 2-15x above the lower bound, so the sketch, not compounding across layers, dominates the error. Oversampling (#40) is the larger lever.
• The per-site cost grows with chi**2 times the product of the layer bonds. Thin Trotter layers run at 0.6-1.4x the sequential time up to depth 5, while random MPOs of bond 8 and 16 at depth 4 are 30x and 136x slower. Stack shallow, thin layers; apply anything else pairwise.