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.

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.

Results

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.