Gate 0.8 · G27

The AQFT crossover law at scale

G26 measured where band-limited approximate QFT beats the full QFT for n = 4..10, then honestly reported that an exact classical surrogate reproduced every cell — so the result was a compiler and noise-resilience finding, not a quantum-advantage claim. G27 pushes the same protocol to n = 12, 14 and 16 and measures the surrogate cost per width instead of assuming it.

36 cellsn = 4, 6, 8, 10, 12, 14analytic bound 63/63

Generated 2026-08-24T15:21:42Z. Source of truth: src/data/demos/aqft_crossover_scaled.json, produced by quantum/aqft_law/scale.py.

Crossover surface k*(n, p)

k* is the smallest band width whose measured success probability sits inside the shot envelope of the full-QFT value at the same noise level. Every column below is a separate emulator sweep; the saving is the number of two-qubit gates the truncation removes at no measured cost.

noise	n = 4	n = 6	n = 8	n = 10	n = 12	n = 14
ideal	k*=2/3−2 2q	k*=3/5−6 2q	k*=3/7−20 2q	k*=3/9−42 2q	k*=4/11−56 2q	k*=4/13−90 2q
1x_H2	k*=2/3−2 2q	k*=3/5−6 2q	k*=3/7−20 2q	k*=3/9−42 2q	k*=3/11−72 2q	k*=3/13−110 2q
5x_H2	k*=2/3−2 2q	k*=2/5−12 2q	k*=2/7−30 2q	k*=2/9−56 2q	k*=2/11−90 2q	k*=1/13−156 2q
20x_H2	k*=1/3−6 2q	k*=1/5−20 2q	k*=1/7−42 2q	k*=1/9−72 2q	k*=1/11−110 2q	k*=1/13−156 2q
100x_H2	k*=1/3−6 2q	k*=1/5−20 2q	k*=1/7−42 2q	k*=1/9−72 2q	k*=1/11−110 2q	k*=1/13−156 2q
400x_H2	k*=1/3−6 2q	k*=1/5−20 2q	k*=1/7−42 2q	k*=1/9−72 2q	k*=1/11−110 2q	k*=1/13−156 2q

Max gate saving across the surface: 156 two-qubit gates. Strict wins (truncated band strictly beats the full QFT): 6.

n	cells	strict wins	noise levels
4	6	0 (0.0%)	—
6	6	0 (0.0%)	—
8	6	0 (0.0%)	—
10	6	2 (33.3%)	5x_H2, 20x_H2
12	6	2 (33.3%)	5x_H2, 20x_H2
14	6	2 (33.3%)	1x_H2, 5x_H2

The strict-win region appears at n = 10 and holds at n = 12 — two of six noise levels at each width. On the widths measured so far it does not grow with n; it is a high-noise effect, not a width effect. n = 14 is mid-sweep locally; n = 16 is not run on Selene at all (a noisy density-matrix cell at that width is unaffordable in the sandbox) and is instead measured on the Nexus cloud emulator lane below.

n = 16 cloud leg — H2-Emulator

The same protocol — phase-state prep, band-limited inverse QFT, bit reversal, target j = 1, 512 shots — run at n = 16 on the Nexus H2-Emulator lane, where physical error rate is swept with error_params.scale (verified tunable by quantum.aqft_law.nexus_probe). This is a cloud emulator lane, not the QPU.

1 live cellsbilled 0 HQCscale 1× · k* = 10

K	2q	p ideal	p @ 1×
10	234	1	0.9355Δ0.0645

Source of truth: src/data/demos/aqft_crossover_n16_nexus.json, produced by quantum/aqft_law/nexus_scale_leg.py. Shot envelope 4·√(0.5/512) = 0.125. Rows are recorded as measured; the bands still running are absent rather than interpolated. Every submitted job is listed in the saved-Ref ledger.

Analytic truncation bound, re-certified at every width

A bound that only holds at the widths it was written against is not a law. The Coppersmith dropped-angle certificate |p_K - p_full| <= 2 * sum_{d>K} 2*pi/2^d is re-checked against the exact oracle at each width; tightness is the ratio of the observed deviation to the certified bound.

n	cells	passing	max tightness	mean tightness
4	3	3/3	0	0
6	5	5/5	0.0061	0.0031
8	7	7/7	0.0024	0.0013
10	9	9/9	0.0013	0.0006
12	11	11/11	0.0013	0.0004
14	13	13/13	0.0006	0.0002
16	15	15/15	0.0006	0.0001

The bound holds everywhere and is loose by two to three orders of magnitude — it certifies safety, it does not predict the deviation.

What this is not

Emulator only (Selene). No hardware shots — L1 receipts plus part of L2, never L3.
The depolarizing model is a scaled stand-in for H-series noise, not a device-calibrated model.
Wider n does not make the ideal distribution classically hard; see the dequantization block.
k* is measured on this circuit family and target set; it is not proved for arbitrary inputs.
Emulator lanes only. No hardware shots were purchased for this gate.

verdict verifiedPASS4/4 criteria

Each criterion below is recomputed from the committed rows by quantum/verdicts.py; this chip reads the result rather than restating it.

passideal_cells_match_oracle
every ideal-noise cell agrees with the exact statevector value inside its 4-sigma envelope
measured 0.0215 · threshold 0.0722
passstrict_wins_recomputed
the strict-win count recomputes from the committed rows
measured 6 · threshold 6
passtruncation_saves_gates_everywhere
in every cell some truncated band uses fewer two-qubit gates than the full QFT while staying inside the envelope of the best row
measured 36 · threshold 36
passcell_count_matches_grid
the committed cell count matches the stated grid
measured 36 · threshold 36