A finite-shot mean-squared-error boundary for Richardson zero-noise extrapolation

arXiv:2605.08251v2 Announce Type: replace Abstract: Zero-noise extrapolation (ZNE) reduces the bias of noisy quantum-circuit measurements, but Richardson coefficients and shot splitting increase their statistical error. This paper determines the noise strength at which Richardson ZNE starts to lower the mean-squared error relative to the unmitigated estimator at a fixed shot budget $B$. The boundary is set by the balance between the leading squared-bias improvement and the leading excess-variance penalty: depending on the variance exponent of the measurement, it shrinks with $B$, becomes a budget threshold, or disappears. For fixed Richardson extrapolation with linear leading bias, the Richardson order and the shot allocation change only constants; the boundary exponent is set by the measured observable and protocol. Deterministic stabilizer measurements and nonzero-variance Hamiltonian or variational-energy measurements have boundaries that shrink as $B^{-1}$ and $B^{-1/2}$, and for finite-outcome shot measurements these two classes are the only ones. Qiskit Aer simulations of Greenberger-Horne-Zeilinger (GHZ) and quantum approximate optimization algorithm (QAOA) circuits reproduce this separation, and on IBM Quantum hardware ZNE lowers the mean-squared error at equal budget in 75 of 101 instances.

Abstract:Zero-noise extrapolation (ZNE) reduces the bias of noisy quantum-circuit measurements, but Richardson coefficients and shot splitting increase their statistical error. This paper determines the noise strength at which Richardson ZNE starts to lower the mean-squared error relative to the unmitigated estimator at a fixed shot budget $B$. The boundary is set by the balance between the leading squared-bias improvement and the leading excess-variance penalty: depending on the variance exponent of the measurement, it shrinks with $B$, becomes a budget threshold, or disappears. For fixed Richardson extrapolation with linear leading bias, the Richardson order and the shot allocation change only constants; the boundary exponent is set by the measured observable and protocol. Deterministic stabilizer measurements and nonzero-variance Hamiltonian or variational-energy measurements have boundaries that shrink as $B^{-1}$ and $B^{-1/2}$, and for finite-outcome shot measurements these two classes are the only ones. Qiskit Aer simulations of Greenberger-Horne-Zeilinger (GHZ) and quantum approximate optimization algorithm (QAOA) circuits reproduce this separation, and on IBM Quantum hardware ZNE lowers the mean-squared error at equal budget in 75 of 101 instances.

26 pages, 9 figures, 10 tables. v2: revised and extended version with a new title; restructured theory (general local theorem, fixed-Richardson derivation, observable classes), full simulation details, equal-budget analysis of all IBM hardware runs, and a supercritical synthetic control. Data and code: Zenodo, DOI https://doi.org/10.5281/zenodo.23405717

arXiv:2605.08251 [quant-ph]

(or arXiv:2605.08251v2 [quant-ph] for this version)

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A finite-shot mean-squared-error boundary for Richardson zero-noise extrapolation

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