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On the success probability of quantum order finding

Martin Ekerå

TL;DR

This work provides a formal, quantitative lower bound on the probability that Shor's quantum order-finding algorithm recovers the true order $r$ in a single run, without increasing the quantum resource length. By incorporating two constrained searches in the classical post-processing—one near the observed frequency and a second via continued fractions or lattice-based methods—the authors guarantee a high success probability for any $r$, with the bound approaching 1 as $r$ grows. They also establish corollaries showing that, under the same framework, one can factor any integer $N$ completely in a single order-finding run with high probability, and extend these results to the case where the factorization of $N$ is recovered via $r$ once $g$ is chosen uniformly at random from $\mathbb{Z}_N^*$. The analysis hinges on precise approximations of the frequency distribution $P(\alpha_r)$ by $\widetilde{P}(\alpha_r)$, proving approximate uniformity, and leveraging cm-smoothness of the gcd $d=\gcd(r,z)$ to recover $r$ from $\tilde{r}=r/d$, all while maintaining polynomial-time classical post-processing. This strengthens the practical viability of Shor-type order finding for factoring tasks, especially under realistic constraints on quantum resources.

Abstract

We prove a lower bound on the probability of Shor's order-finding algorithm successfully recovering the order $r$ in a single run. The bound implies that by performing two limited searches in the classical post-processing part of the algorithm, a high success probability can be guaranteed, for any $r$, without re-running the quantum part or increasing the exponent length compared to Shor. Asymptotically, in the limit as $r$ tends to infinity, the probability of successfully recovering $r$ in a single run tends to one. Already for moderate $r$, a high success probability exceeding e.g. $1 - 10^{-4}$ can be guaranteed. As corollaries, we prove analogous results for the probability of completely factoring any integer $N$ in a single run of the order-finding algorithm.

On the success probability of quantum order finding

TL;DR

This work provides a formal, quantitative lower bound on the probability that Shor's quantum order-finding algorithm recovers the true order in a single run, without increasing the quantum resource length. By incorporating two constrained searches in the classical post-processing—one near the observed frequency and a second via continued fractions or lattice-based methods—the authors guarantee a high success probability for any , with the bound approaching 1 as grows. They also establish corollaries showing that, under the same framework, one can factor any integer completely in a single order-finding run with high probability, and extend these results to the case where the factorization of is recovered via once is chosen uniformly at random from . The analysis hinges on precise approximations of the frequency distribution by , proving approximate uniformity, and leveraging cm-smoothness of the gcd to recover from , all while maintaining polynomial-time classical post-processing. This strengthens the practical viability of Shor-type order finding for factoring tasks, especially under realistic constraints on quantum resources.

Abstract

We prove a lower bound on the probability of Shor's order-finding algorithm successfully recovering the order in a single run. The bound implies that by performing two limited searches in the classical post-processing part of the algorithm, a high success probability can be guaranteed, for any , without re-running the quantum part or increasing the exponent length compared to Shor. Asymptotically, in the limit as tends to infinity, the probability of successfully recovering in a single run tends to one. Already for moderate , a high success probability exceeding e.g. can be guaranteed. As corollaries, we prove analogous results for the probability of completely factoring any integer in a single run of the order-finding algorithm.
Paper Structure (62 sections, 28 theorems, 124 equations, 1 table)

This paper contains 62 sections, 28 theorems, 124 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quantum order finding
  4. Understanding the probability distribution
  5. Shor's original analysis and post-processing
  6. Earlier works that follow up on Shor's work
  7. Precise simulations and estimates
  8. Overview of our contribution
  9. Overview of our analysis
  10. On the relation to our recent work on factoring completely
  11. Approximating $P(\alpha_r)$ by $\widetilde{P}(\alpha_r)$
  12. Supporting claims
  13. Approximating $P(\alpha_r)$ by $T(\alpha_r)$
  14. Approximating $T(\alpha_r)$ by $\widetilde{P}(\alpha_r)$
  15. Approximating $P(\alpha_r)$ by $\widetilde{P}(\alpha_r)$
  16. ...and 47 more sections

Key Result

Lemma 1

The error when approximating $P(\alpha_r)$ by $T(\alpha_r)$ is bounded by

Theorems & Definitions (86)

  • Claim 1
  • Claim 2
  • Claim 3
  • Claim 4
  • Claim 5
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 76 more