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  • Open Access

Quantum period finding using one-qubit reduced density matrices

Marco Bernardi*

  • *Contact author: bmarco@caltech.edu
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Phys. Rev. Research 8, 023137 – Published 8 May, 2026

DOI: https://doi.org/10.1103/bcxp-1qld

Abstract

The quantum period-finding (QPF) algorithm can compute the period of a function exponentially faster than the best-known classical algorithm. In standard QPF, the output state has a primary contribution from r high-probability bit strings, where r is the period. Measurement of this state, combined with continued fraction analysis, reveals the unknown period. Here, we explore a different approach to QPF, where the period is obtained from single-qubit quantities—specifically, the set of one-qubit reduced density matrices (1-RDMs)—rather than the output bit strings of the entire quantum circuit. Using state-vector simulations, we compute the 1-RDMs of the QPF circuit for a generic periodic function. Analysis of these 1-RDMs as a function of period reveals distinctive patterns, which allows us to obtain the unknown period from the 1-RDMs using a numerical root-finding approach. Our results show that the 1-RDMs—a set of O(n) one-qubit marginals—contain enough information to reconstruct the period, which is typically obtained by sampling the space of O(2n) bit strings. Conceptually, this can be viewed as a “compression” of the information in the QPF algorithm, which enables period finding from n one-qubit marginals. Our results motivate the development of approximate simulations of reduced density matrices to design period-finding algorithms.

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Outline

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Phys. Rev. Research 8, 023137– Published 8 May, 2026
  • Received 17 November 2025
  • Accepted 13 April 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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