Stability of the quantum phase estimation algorithm under uniform distribution of eigenvalues

Gordeichuk M.O., Kiselev O.M.
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This paper establishes quantitative conditions for the stability of the Quantum Phase Estimation (QPE) algorithm under the assumption of a uniform distribution of eigenvalues of the unitary operator. Using perturbation theory for linear operators, we demonstrate that the accuracy of phase estimation is fundamentally limited by a logarithmic dependence on the perturbation magnitude: the number of reliably recoverable binary digits of the phase satisfies the condition $n=o(-\log_2^{}(\epsilon))$. Furthermore, we show that distinct phases remain resolvable only if the perturbation does not exceed the minimal distance $\frac{1}{m}$ between adjacent phases, which leads to the condition $m=o\left(\epsilon^{-1}\right)$. These results reveal fundamental limitations on the resolving power of QPE in the presence of imperfect input data and are of direct practical relevance for the design of robust quantum algorithms that employ QPE as a~subroutine.

Keywords: quantum, eigevalues, phase esimation, perturbation
Citation in English: Gordeichuk M.O., Kiselev O.M. Stability of the quantum phase estimation algorithm under uniform distribution of eigenvalues // Computer Research and Modeling, 2026, vol. 18, no. 1, pp. 9-24
Citation in English: Gordeichuk M.O., Kiselev O.M. Stability of the quantum phase estimation algorithm under uniform distribution of eigenvalues // Computer Research and Modeling, 2026, vol. 18, no. 1, pp. 9-24
DOI: 10.20537/2076-7633-2026-18-1-9-24
URL: http://crm-en.ics.org.ru/journal/article/3685/
Creative Commons License This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License.

Copyright © 2026 Gordeichuk M.O., Kiselev O.M.

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