The central result
A preprint reports a theorem for a depth-tunable, ancilla-free quantum amplitude-estimation family. It gives near-optimal query–depth scaling up to logarithmic factors and an angle-error guarantee with probability at least 1 − δ for every Grover angle λ from 0 to π/2, when the depth window meets the theorem’s conditions.
That coverage includes both ends of the Grover-angle interval. In the corresponding amplitude conversion, the reported endpoint bound at a = 0 or a = 1 reduces the error limit to the squared-epsilon form, ε².
How the estimate is built
The method, called Windowed Least Squares Amplitude Estimation, samples circuit depths from a discrete window. It records outcomes written as +1 or −1 and estimates the angle by minimizing a mean-square loss.
The proof does not accept every possible depth schedule. It requires an admissible window: for each depth up to T, the combined probability assigned to its positive and negative versions must be at least b/T, with T = max(1, M/σ).
A dial between resource regimes
A corollary describes a continuous interpolation over β from 0 to 1. At β = 1, the method corresponds to classical sampling; at β = 0, it reaches the Heisenberg limit. The reported query and depth orders depend on β, and the construction uses no ancilla qubits or controlled Grover operations.
The loss is evaluated as a discrete cosine polynomial using a discrete cosine transform. The reported classical cost is O(N + K log K), compared with O(NK) for a naive calculation.
The windows did not perform alike
Numerical tests compared the window choices at a fixed query budget and fixed λ = 0.5 over 104 independent runs. At maximum depth M = 1024, the uniform window’s angle error was about 40% lower than the Gaussian window’s and 17% lower than the Kaiser window’s; linear and cubic windows improved on uniform by a further 5% and 10%, respectively.
A separate sweep with the uniform window varied λ over three orders of magnitude at a fixed query budget. Its error curves did not deteriorate as the angle approached the boundary; they moved downward with increasing depth in proportion to an inverse-square-root pattern in M.
Shot counts also varied by window. Relative to the uniform window at M = 1024, Gaussian and Kaiser required 3.1 and 1.5 times as many shots, while linear and cubic required 0.75 and 0.63 times as many.
Conditions and caveats
The theorem’s uniform guarantee is conditional: it applies to admissible windows, and the reported resource scaling includes logarithmic factors with hidden constants that depend on the admissibility parameters.
The Cramér–Rao curve used in the numerical discussion is only a local benchmark away from the boundary. It is not a rigorous lower bound for this least-squares estimator because it assumes regular unbiased estimation and ignores global aliases.
The evidence is algorithmic and numerical: the interpolation is a resource result, and the window comparisons use fixed-budget simulations rather than a physical-device test.
Publication and disclosures
The work is identified as arXiv version 1 dated 25 Aug 2026. It reports an industrial PhD Studentship funded by Moderna and Quantum Motion, UKRI Future Leaders Fellowship support under MR/Y015843/1 and additional UKRI funding. The numerical code is reported as available at the stated GitHub repository, and the paper discloses Claude Opus 5 assistance with the numerics and some text.
Paper data and sources
Original title: Nearly Optimal Amplitude Estimation at any Depth
Authors: Jona Erle, Bálint Koczor
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text