Gate G26 · Crossover law

AQFT crossover law k*(n, p)

The approximate QFT drops every controlled-phase rotation beyond a band width K. In an ideal machine that is a pure loss: each dropped rotation costs a little accuracy. On a noisy machine each rotation also costs a two-qubit gate, and gates are the thing that is actually breaking. So there is a band width k* below which truncation stops being a compromise and starts being the better circuit. This gate measures that surface directly: 24 emulator cells across 4, 6, 8, 10 qubits and a six-step depolarizing ladder, with an exact statevector oracle at every point so truncation error and noise are never confused with one another.

crossover law verified24 cells24/24 save gates for free72 two-qubit gates saved at bestemulator only · no HQCs

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Reproduce on your laptop

The entire G26 pipeline is runnable locally with the pinned dependency set in quantum/requirements.txt. The cross-platform corroboration script is quantum/gate26_cross_platform/run_all.py; it accepts a free Nexus account for the hosted emulator lanes and falls back to Selene + NumPy for fully local runs.

How k* is defined

For each (n, noise) cell we run every band width K from 1 up to the full QFT, measure P(measured = j) over fixed targets, and take k* as the smallest K whose measured success probability is within the statistical envelope of the full QFT. Everything above k* is paying two-qubit gates for accuracy the shot noise cannot see. The envelope is 4*sqrt(0.5/(shots*|targets|)), the same 4σ binomial rule used across the rest of this site.

Shots per target: 512
Seed: 7
Oracle: exact statevector
Noise model: depolarizing · H2-scaled

The H2 baseline the ladder multiplies: p_2q = 0.00129, p_1q = 0.000129, p_meas = 0.00135.

The crossover surface

k* per (n, noise) cell, with the two-qubit gates saved at that band width underneath. k* shrinks monotonically as noise rises — the noisier the machine, the narrower the QFT it wants.

n	ideal	1x_H2	5x_H2	20x_H2	100x_H2	400x_H2
4	k* = 2/3 −2 2q gates	k* = 2/3 −2 2q gates	k* = 2/3 −2 2q gates	k* = 1/3 −6 2q gates	k* = 1/3 −6 2q gates	k* = 1/3 −6 2q gates
6	k* = 3/5 −6 2q gates	k* = 3/5 −6 2q gates	k* = 2/5 −12 2q gates	k* = 1/5 −20 2q gates	k* = 1/5 −20 2q gates	k* = 1/5 −20 2q gates
8	k* = 3/7 −20 2q gates	k* = 3/7 −20 2q gates	k* = 2/7 −30 2q gates	k* = 1/7 −42 2q gates	k* = 1/7 −42 2q gates	k* = 1/7 −42 2q gates
10	k* = 3/9 −42 2q gates	k* = 3/9 −42 2q gates	k* = 2/9 −56 2q gates truncation strictly wins	k* = 1/9 −72 2q gates truncation strictly wins	k* = 1/9 −72 2q gates	k* = 1/9 −72 2q gates

Where truncation strictly wins

In 2 of 24 cells the best band width is not just as good as the full QFT, it beats it outright by more than the envelope — the dropped rotations were costing more in noise than they were worth in phase resolution.

n = 10 · 5x_H2

best K = 3 at 55.9%, full QFT at 46.4%

56 two-qubit gates removed at k* = 2.

n = 10 · 20x_H2

best K = 2 at 13.0%, full QFT at 4.6%

72 two-qubit gates removed at k* = 1.

Every cell, band by band

Measured success probability against the exact oracle value for each band width. The gap between them is the noise penalty; the gap between the oracle value and 1.0 is the truncation error. Keeping the two separate is the whole point of the oracle leg.

n = 4 · ideal

k* = 2 · targets 1, 5, 15 · 512 shots each · envelope ±0.0722

K	2q gates	measured	oracle	noise penalty
1	12	75.1%	74.4%	-0.0066
2k*	16	96.4%	96.2%	-0.0016
3full QFT	18	100.0%	100.0%	0.0000

n = 4 · 1x_H2

k* = 2 · targets 1, 5, 15 · 512 shots each · envelope ±0.0722

K	2q gates	measured	oracle	noise penalty
1	12	71.0%	74.4%	0.0338
2k*	16	93.2%	96.2%	0.0303
3full QFT	18	95.5%	100.0%	0.0449

n = 4 · 5x_H2

k* = 2 · targets 1, 5, 15 · 512 shots each · envelope ±0.0722

K	2q gates	measured	oracle	noise penalty
1	12	67.6%	74.4%	0.0683
2k*	16	83.6%	96.2%	0.1260
3full QFT	18	84.8%	100.0%	0.1523

n = 4 · 20x_H2

k* = 1 · targets 1, 5, 15 · 512 shots each · envelope ±0.0722

K	2q gates	measured	oracle	noise penalty
1k*	12	48.4%	74.4%	0.2604
2	16	54.3%	96.2%	0.4190
3full QFT	18	53.8%	100.0%	0.4622

n = 4 · 100x_H2

k* = 1 · targets 1, 5, 15 · 512 shots each · envelope ±0.0722

K	2q gates	measured	oracle	noise penalty
1k*	12	14.3%	74.4%	0.6015
2	16	10.8%	96.2%	0.8539
3full QFT	18	10.2%	100.0%	0.8984

n = 4 · 400x_H2

k* = 1 · targets 1, 5, 15 · 512 shots each · envelope ±0.0722

K	2q gates	measured	oracle	noise penalty
1k*	12	5.9%	74.4%	0.6848
2	16	5.9%	96.2%	0.9033
3full QFT	18	6.0%	100.0%	0.9401

n = 6 · ideal

k* = 3 · targets 1, 21, 63 · 512 shots each · envelope ±0.0722

K	2q gates	measured	oracle	noise penalty
1	19	54.1%	53.5%	-0.0058
2	27	86.6%	87.6%	0.0100
3k*	33	98.4%	98.2%	-0.0027
4	37	99.8%	99.8%	-0.0005
5full QFT	39	100.0%	100.0%	0.0000

n = 6 · 1x_H2

k* = 3 · targets 1, 21, 63 · 512 shots each · envelope ±0.0722

K	2q gates	measured	oracle	noise penalty
1	19	51.8%	53.5%	0.0177
2	27	83.0%	87.6%	0.0458
3k*	33	92.8%	98.2%	0.0540
4	37	92.2%	99.8%	0.0757
5full QFT	39	93.0%	100.0%	0.0697

n = 6 · 5x_H2

k* = 2 · targets 1, 21, 63 · 512 shots each · envelope ±0.0722

K	2q gates	measured	oracle	noise penalty
1	19	47.3%	53.5%	0.0626
2k*	27	67.1%	87.6%	0.2053
3	33	74.7%	98.2%	0.2350
4	37	73.2%	99.8%	0.2652
5full QFT	39	72.1%	100.0%	0.2793

n = 6 · 20x_H2

k* = 1 · targets 1, 21, 63 · 512 shots each · envelope ±0.0722

K	2q gates	measured	oracle	noise penalty
1k*	19	27.6%	53.5%	0.2592
2	27	32.3%	87.6%	0.5530
3	33	31.4%	98.2%	0.6679
4	37	28.8%	99.8%	0.7092
5full QFT	39	27.0%	100.0%	0.7298

n = 6 · 100x_H2

k* = 1 · targets 1, 21, 63 · 512 shots each · envelope ±0.0722

K	2q gates	measured	oracle	noise penalty
1k*	19	3.1%	53.5%	0.5046
2	27	2.5%	87.6%	0.8505
3	33	2.2%	98.2%	0.9596
4	37	2.1%	99.8%	0.9768
5full QFT	39	2.2%	100.0%	0.9779

n = 6 · 400x_H2

k* = 1 · targets 1, 21, 63 · 512 shots each · envelope ±0.0722

K	2q gates	measured	oracle	noise penalty
1k*	19	1.7%	53.5%	0.5183
2	27	1.5%	87.6%	0.8609
3	33	1.5%	98.2%	0.9667
4	37	1.5%	99.8%	0.9826
5full QFT	39	1.4%	100.0%	0.9857

n = 8 · ideal

k* = 3 · targets 1, 85, 255 · 512 shots each · envelope ±0.0722

K	2q gates	measured	oracle	noise penalty
1	26	41.9%	42.9%	0.0107
2	38	78.9%	78.7%	-0.0019
3k*	48	95.6%	95.5%	-0.0004
4	56	99.2%	99.2%	-0.0005
5	62	99.9%	99.9%	0.0001
6	66	100.0%	100.0%	-0.0002
7full QFT	68	100.0%	100.0%	0.0000

n = 8 · 1x_H2

k* = 3 · targets 1, 85, 255 · 512 shots each · envelope ±0.0722

K	2q gates	measured	oracle	noise penalty
1	26	41.3%	42.9%	0.0159
2	38	73.4%	78.7%	0.0528
3k*	48	86.5%	95.5%	0.0901
4	56	87.8%	99.2%	0.1134
5	62	89.0%	99.9%	0.1089
6	66	89.2%	100.0%	0.1079
7full QFT	68	88.3%	100.0%	0.1165

n = 8 · 5x_H2

k* = 2 · targets 1, 85, 255 · 512 shots each · envelope ±0.0722

K	2q gates	measured	oracle	noise penalty
1	26	31.9%	42.9%	0.1103
2k*	38	56.6%	78.7%	0.2207
3	48	60.4%	95.5%	0.3518
4	56	62.8%	99.2%	0.3634
5	62	61.7%	99.9%	0.3817
6	66	59.4%	100.0%	0.4054
7full QFT	68	60.6%	100.0%	0.3945

n = 8 · 20x_H2

k* = 1 · targets 1, 85, 255 · 512 shots each · envelope ±0.0722

K	2q gates	measured	oracle	noise penalty
1k*	26	16.2%	42.9%	0.2679
2	38	20.7%	78.7%	0.5801
3	48	19.2%	95.5%	0.7633
4	56	16.0%	99.2%	0.8315
5	62	13.7%	99.9%	0.8621
6	66	12.6%	100.0%	0.8735
7full QFT	68	14.8%	100.0%	0.8522

n = 8 · 100x_H2

k* = 1 · targets 1, 85, 255 · 512 shots each · envelope ±0.0722

K	2q gates	measured	oracle	noise penalty
1k*	26	1.3%	42.9%	0.4163
2	38	0.5%	78.7%	0.7826
3	48	0.8%	95.5%	0.9475
4	56	0.5%	99.2%	0.9864
5	62	0.2%	99.9%	0.9969
6	66	0.5%	100.0%	0.9953
7full QFT	68	0.3%	100.0%	0.9967

n = 8 · 400x_H2

k* = 1 · targets 1, 85, 255 · 512 shots each · envelope ±0.0722

K	2q gates	measured	oracle	noise penalty
1k*	26	0.4%	42.9%	0.4254
2	38	0.7%	78.7%	0.7800
3	48	0.7%	95.5%	0.9488
4	56	0.3%	99.2%	0.9884
5	62	0.7%	99.9%	0.9923
6	66	0.6%	100.0%	0.9940
7full QFT	68	0.3%	100.0%	0.9967

n = 10 · ideal

k* = 3 · targets 1, 341, 1023 · 512 shots each · envelope ±0.0722

K	2q gates	measured	oracle	noise penalty
1	33	36.9%	37.4%	0.0050
2	49	69.8%	71.3%	0.0147
3k*	63	94.1%	92.7%	-0.0137
4	75	98.3%	98.4%	0.0012
5	85	99.9%	99.7%	-0.0016
6	93	100.0%	100.0%	-0.0005
7	99	100.0%	100.0%	-0.0001
8	103	100.0%	100.0%	0.0000
9full QFT	105	100.0%	100.0%	0.0000

n = 10 · 1x_H2

k* = 3 · targets 1, 341, 1023 · 512 shots each · envelope ±0.0722

K	2q gates	measured	oracle	noise penalty
1	33	35.1%	37.4%	0.0232
2	49	65.4%	71.3%	0.0590
3k*	63	82.0%	92.7%	0.1067
4	75	87.0%	98.4%	0.1145
5	85	84.9%	99.7%	0.1482
6	93	84.9%	100.0%	0.1505
7	99	84.2%	100.0%	0.1575
8	103	84.1%	100.0%	0.1588
9full QFT	105	83.3%	100.0%	0.1673

n = 10 · 5x_H2

k* = 2 · targets 1, 341, 1023 · 512 shots each · envelope ±0.0722

K	2q gates	measured	oracle	noise penalty
1	33	26.8%	37.4%	0.1059
2k*	49	46.3%	71.3%	0.2497
3	63	55.9%	92.7%	0.3685
4	75	52.7%	98.4%	0.4576
5	85	49.7%	99.7%	0.4997
6	93	47.9%	100.0%	0.5203
7	99	46.9%	100.0%	0.5312
8	103	45.6%	100.0%	0.5436
9full QFT	105	46.4%	100.0%	0.5365

n = 10 · 20x_H2

k* = 1 · targets 1, 341, 1023 · 512 shots each · envelope ±0.0722

K	2q gates	measured	oracle	noise penalty
1k*	33	11.1%	37.4%	0.2628
2	49	13.0%	71.3%	0.5824
3	63	11.7%	92.7%	0.8099
4	75	8.9%	98.4%	0.8951
5	85	6.2%	99.7%	0.9353
6	93	6.5%	100.0%	0.9350
7	99	5.2%	100.0%	0.9478
8	103	4.9%	100.0%	0.9512
9full QFT	105	4.6%	100.0%	0.9538

n = 10 · 100x_H2

k* = 1 · targets 1, 341, 1023 · 512 shots each · envelope ±0.0722

K	2q gates	measured	oracle	noise penalty
1k*	33	0.5%	37.4%	0.3689
2	49	0.2%	71.3%	0.7107
3	63	0.0%	92.7%	0.9271
4	75	0.3%	98.4%	0.9816
5	85	0.1%	99.7%	0.9958
6	93	0.0%	100.0%	0.9995
7	99	0.2%	100.0%	0.9980
8	103	0.0%	100.0%	1.0000
9full QFT	105	0.1%	100.0%	0.9987

n = 10 · 400x_H2

k* = 1 · targets 1, 341, 1023 · 512 shots each · envelope ±0.0722

K	2q gates	measured	oracle	noise penalty
1k*	33	0.1%	37.4%	0.3728
2	49	0.1%	71.3%	0.7120
3	63	0.1%	92.7%	0.9264
4	75	0.1%	98.4%	0.9829
5	85	0.0%	99.7%	0.9971
6	93	0.1%	100.0%	0.9982
7	99	0.1%	100.0%	0.9993
8	103	0.0%	100.0%	1.0000
9full QFT	105	0.1%	100.0%	0.9987

Second compiler: does the saving survive TKET?

A two-qubit saving that a production optimiser can also find inside the full QFT is not a saving. Every band width was rebuilt as a pytket circuit — gate for gate against the Guppy kernel, including the 2-CX controlled-phase form and the 3-CX bit reversal — and compiled offline against the H2-2 device model at optimisation levels 0, 1, 2. Each compiled form is gated on a global-phase-free unitary oracle at 1e-9 before its gate count is allowed into the table below.

72/72 unitary-equivalent20/20 savings survive compilationtket_confirms_truncation_saving

At level 2 TKET roughly halves both circuits by merging the CX pairs into native ZZPhase rotations, so the absolute saving shrinks — but it never disappears: the worst-case retention is 50% of the source-level saving, and the largest surviving gap is 36 two-qubit gates. Truncation buys depth the compiler cannot recover on its own.

n	K	source 2q	TKET 2q	source saving	TKET saving
4	1	12	3	6	3
4	2	16	5	2	1
6	2	27	9	12	6
6	3	33	12	6	3
6	4	37	14	2	1
8	4	56	22	12	6
8	5	62	25	6	3
8	6	66	27	2	1
10	6	93	39	12	6
10	7	99	42	6	3
10	8	103	44	2	1

Rows shown are the widest four bands per qubit count, at optimisation level 2, versus that n's full QFT. Compilation is fully offline: QuantinuumAPIOffline, no credentials, no HQCs.

Third engine: the Nexus cloud lanes

A reduced slice of the surface (n = 6, target = 1, 512 shots, seed 7) was executed on H2-1LE, H2-Emulator, Helios-1E-lite through Quantinuum Nexus and compared against the exact statevector oracle inside the 4·√(0.5/shots) = 0.125 envelope.

15/15 cells within envelope15 live jobs0 HQC billed

device	K	2q	p ideal	p measured	\|Δ\|	verdict
H2-1LE	1	19	0.8112	0.8184	0.0072	PASS
H2-1LE	2	27	0.9504	0.9434	0.007	PASS
H2-1LE	3	33	0.988	0.9844	0.0036	PASS
H2-1LE	4	37	0.9976	1	0.0024	PASS
H2-1LE	5	39	1	1	0	PASS
H2-Emulator	1	19	0.8112	0.7852	0.026	PASS
H2-Emulator	2	27	0.9504	0.9414	0.009	PASS
H2-Emulator	3	33	0.988	0.9668	0.0212	PASS
H2-Emulator	4	37	0.9976	0.9785	0.0191	PASS
H2-Emulator	5	39	1	0.9922	0.0078	PASS
Helios-1E-lite	1	19	0.8112	0.7949	0.0163	PASS
Helios-1E-lite	2	27	0.9504	0.9102	0.0402	PASS
Helios-1E-lite	3	33	0.988	0.959	0.029	PASS
Helios-1E-lite	4	37	0.9976	0.9668	0.0308	PASS
Helios-1E-lite	5	39	1	0.957	0.043	PASS

Every job carries its own meter (device, job id, shots, seed, estimated and billed HQC) in the dump's nexus_leg block. These are emulator lanes, not hardware: no QPU shots were taken and nothing was billed.

Cross-platform summary

The same n = 6, target = 1, 512 shot slice was executed across multiple emulators and engines. Each row is checked against the same exact-oracle envelope 4·√(0.5/shots) = 0.125.

25/25 cells pass25 live jobs0 HQC billed5 engines

engine	device	K	2q	p ideal	p measured	\|Δ\|	verdict
aer	aer_simulator	1	19	0.8112	0.7832	0.0280	PASS
aer	aer_simulator	2	27	0.9504	0.9414	0.0090	PASS
aer	aer_simulator	3	33	0.9880	0.9883	0.0003	PASS
aer	aer_simulator	4	37	0.9976	1.0000	0.0024	PASS
aer	aer_simulator	5	39	1.0000	1.0000	0.0000	PASS
h-series	H2-1LE	1	19	0.8112	0.7988	0.0124	PASS
h-series	H2-1LE	2	27	0.9504	0.9434	0.0070	PASS
h-series	H2-1LE	3	33	0.9880	0.9863	0.0017	PASS
h-series	H2-1LE	4	37	0.9976	1.0000	0.0024	PASS
h-series	H2-1LE	5	39	1.0000	1.0000	0.0000	PASS
h-series	H2-Emulator	1	19	0.8112	0.7949	0.0163	PASS
h-series	H2-Emulator	2	27	0.9504	0.9473	0.0031	PASS
h-series	H2-Emulator	3	33	0.9880	0.9785	0.0095	PASS
h-series	H2-Emulator	4	37	0.9976	0.9785	0.0191	PASS
h-series	H2-Emulator	5	39	1.0000	0.9844	0.0156	PASS
helios	Helios-1E-lite	1	19	0.8112	0.7852	0.0260	PASS
helios	Helios-1E-lite	2	27	0.9504	0.9473	0.0031	PASS
helios	Helios-1E-lite	3	33	0.9880	0.9551	0.0329	PASS
helios	Helios-1E-lite	4	37	0.9976	0.9727	0.0249	PASS
helios	Helios-1E-lite	5	39	1.0000	0.9688	0.0312	PASS
qulacs	Qulacs	1	19	0.8112	0.8223	0.0111	PASS
qulacs	Qulacs	2	27	0.9504	0.9648	0.0144	PASS
qulacs	Qulacs	3	33	0.9880	0.9922	0.0042	PASS
qulacs	Qulacs	4	37	0.9976	0.9980	0.0004	PASS
qulacs	Qulacs	5	39	1.0000	1.0000	0.0000	PASS

Engine families are the Quantinuum Nexus config classes used to submit each job (Aer, H2, Helios, Qulacs, Selene). Meters for estimated and billed HQC are recorded in the artifact for every live job.

Batch integrity: the Bell control

Five engines agreeing proves the engines are consistent, not that the batch decoded correctly. A known-fidelity |Φ+⟩ pair was submitted into every batch and accepted when the anti-correlated fraction stays inside 4·√(0.5/shots). A failed control fails its whole batch.

batch integrity PASS

device	shots	anti-correlated	envelope	verdict
H2-1LE	512	0 (0)	0.125	PASS
H2-Emulator	512	3 (0.0059)	0.125	PASS
Helios-1E-lite	512	2 (0.0039)	0.125	PASS
aer_simulator	512	0 (0)	0.125	PASS
Qulacs	512	0 (0)	0.125	PASS

Dequantization gate

DEQUANTIZED — a classical surrogate reproduces every engine row inside the envelope. G26 is a compiler and noise-resilience result, not a quantum-advantage claim.

Surrogate: exact statevector Born probabilities, multinomial-sampled at the same shot count and seed as the engine lanes. Scored against the same envelope 0.125.

K	p classical	p ideal	max \|Δ\| vs engines	reproduced
1	0.834	0.8112	0.0508	yes
2	0.9434	0.9504	0.0214	yes
3	0.9883	0.988	0.0332	yes
4	0.998	0.9976	0.0253	yes
5	1	1	0.0312	yes

Shot ladder

The same five K values re-run on H2-1LE at 128 / 512 / 2048 shots. A metric that stays flat while the envelope shrinks is shot-noise-limited, not an artefact of the 512-shot choice.

15/15 ladder cells pass

shots	K	p ideal	p measured	\|Δ\|	envelope	verdict
128	1	0.8112	0.8516	0.0404	0.25	PASS
128	2	0.9504	0.9766	0.0262	0.25	PASS
128	3	0.988	0.9844	0.0036	0.25	PASS
128	4	0.9976	1	0.0024	0.25	PASS
128	5	1	1	0	0.25	PASS
512	1	0.8112	0.8203	0.0091	0.125	PASS
512	2	0.9504	0.9531	0.0027	0.125	PASS
512	3	0.988	0.9902	0.0022	0.125	PASS
512	4	0.9976	1	0.0024	0.125	PASS
512	5	1	1	0	0.125	PASS
2048	1	0.8112	0.8076	0.0036	0.0625	PASS
2048	2	0.9504	0.9502	0.0002	0.0625	PASS
2048	3	0.988	0.9917	0.0037	0.0625	PASS
2048	4	0.9976	0.9971	0.0005	0.0625	PASS
2048	5	1	1	0	0.0625	PASS

Trust layer: L1 receipts (job ids, shots, seed, committed JSON) plus part of L2 (five independent engines and a per-batch Bell control). Not L3: nothing here is cryptographically verified.

What this does not claim

No hardware shots. Emulator only (Selene + Nexus emulator lanes) — L2 on the trust ladder, never L3.
The depolarizing model is a scaled stand-in for H-series noise, not a device-calibrated model.
k* is measured on this circuit family and target set; it is not proved for arbitrary inputs.

The earlier noise-resilience work at the AQFT noise proofs showed a single truncated circuit beating the full QFT on one instance; this gate turns that anecdote into a surface. The second-compiler check lives at G24 and the cloud emulator cross-check at G25.

Generated 2026-08-22T15:15:14Z from quantum/aqft_law/ · results committed at src/data/demos/aqft_crossover_law.json.

n	K	source 2q	TKET 2q	source saving	TKET saving
4	1	12	3	6	3
4	2	16	5	2	1
6	2	27	9	12	6
6	3	33	12	6	3
6	4	37	14	2	1
8	4	56	22	12	6
8	5	62	25	6	3
8	6	66	27	2	1
10	6	93	39	12	6
10	7	99	42	6	3
10	8	103	44	2	1

n	K	source 2q	TKET 2q	source saving	TKET saving
4	1	12	3	6	3
4	2	16	5	2	1
6	2	27	9	12	6
6	3	33	12	6	3
6	4	37	14	2	1
8	4	56	22	12	6
8	5	62	25	6	3
8	6	66	27	2	1
10	6	93	39	12	6
10	7	99	42	6	3
10	8	103	44	2	1

n	K	source 2q	TKET 2q	source saving	TKET saving
4	1	12	3	6	3
4	2	16	5	2	1
6	2	27	9	12	6
6	3	33	12	6	3
6	4	37	14	2	1
8	4	56	22	12	6
8	5	62	25	6	3
8	6	66	27	2	1
10	6	93	39	12	6
10	7	99	42	6	3
10	8	103	44	2	1