The AQFT crossover law k*(n, p)

Abstract

The approximate quantum Fourier transform drops every controlled-phase rotation beyond a band width K. Truncation always costs accuracy in an ideal machine and always saves two-qubit gates in a real one, so there exists a band width k* below which the approximate circuit is the better circuit. We measure that surface directly: 24 cells across n = 4, 6, 8, 10 qubits and a 6-step depolarizing ladder anchored on published H2 error rates, with an exact statevector oracle at every point so approximation error is never confused with noise. In 24 of 24 cells a truncated band matches the full QFT inside the statistical envelope while saving up to 72 two-qubit gates; in 2 cells the truncated circuit is strictly more accurate than the exact one. The claim rests on four independent legs: an exact NumPy oracle, the Selene emulator, an offline pytket compilation against the H2-2 device model, and three Nexus cloud emulator lanes. This is emulator-only evidence (L2 on our trust ladder). No hardware shots were taken and none are claimed.

Method

Leg 1 — the certified truncation bound

|p_K - p_full| <= 2 * sum_{d>K} 2*pi/2^d. Dropping a diagonal rotation of angle θ perturbs the state by at most |θ| in 2-norm and perturbations compose sub-additively, so the sum of dropped angles certifies the worst-case movement of any measurement probability. This is a certificate, not a prediction. Criterion: exact oracle deviation <= certified bound, every (n, K) cell. Result: 24/24 cells satisfy it; the tightest cell reaches 0.626 of the bound, so the measured curve sits well inside it. This is what separates approximation error from noise error in the headline.

Legs 2–4 — independent corroboration

A saving a good optimiser can also find in the full QFT is not a saving; that is why the compiler leg exists. A result one engine produces is not a measurement; that is why the cloud leg exists.

Threats to validity

• The depolarizing ladder is a scaled stand-in for H-series noise, not a device-calibrated model; absolute error rates should not be read off it.
• k* is measured for this circuit family and target set, not proved in general.
• Every engine is an emulator. Agreement between emulators bounds implementation error, not hardware behaviour.
• Shot noise sets the resolution: differences smaller than the envelope are reported as agreement, so k* is a statistical, not exact, threshold.

What is not claimed

• 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 bound is a worst-case certificate, not a prediction of the deviation.
• It bounds approximation error only; noise error is measured, never bounded here.

Reproduce

PYTHONPATH=.pydeps python3 -m quantum.aqft_law.reproduce

Cite

The citation pins a content hash of the dataset. whole document with the provenance block removed, JSON canonicalised (sorted keys, no whitespace). If the hash moves, the dataset moved.

@misc{nadarasa2026aqft,
  author = {Nadarasa, Arun},
  title  = {The AQFT crossover law k*(n, p): a four-leg emulator study},
  year   = {2026},
  note   = {Nadarasa Reduction, gate G26. Artifact src/data/demos/aqft_crossover_law.json,
            sha256 55a663a3ef4232345ed32df3acb7424163d978af2b1d28af7709d9788c7ea46f},
  url    = {https://arunquantum.lovable.app/nadarasa/g26}
}