Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An Efficient Quantum Factoring Algorithm

Oded Regev

TL;DR

This work proposes a quantum factoring scheme that reduces the per-run circuit size to $\\tilde{O}(n^{3/2})$ gates and requires only $\\sqrt{n}+4$ independent executions, followed by polynomial-time classical post-processing. The method uses a multidimensional, Shor-like quantum procedure to sample near dual-lattice vectors, then employs lattice-reduction techniques (LLL) and an extended lattice construction to recover a short vector in $\\mathcal{L}\\setminus\\mathcal{L}_0$, from which a nontrivial factor of the target $N$ is obtained via a gcd computation. The main contributions are the circuit-size improvement, the explicit lattice-based post-processing framework, and the conditional claims based on a number-theoretic heuristic; the practicality depends on whether the heuristic holds and on future optimizations, especially compared to optimized Shor implementations. The result suggests that under plausible assumptions, quantum resources for factoring could be reduced, although real-world performance remains to be demonstrated and verified against existing quantum factoring approaches.

Abstract

We show that $n$-bit integers can be factorized by independently running a quantum circuit with $\tilde{O}(n^{3/2})$ gates for $\sqrt{n}+4$ times, and then using polynomial-time classical post-processing. The correctness of the algorithm relies on a number-theoretic heuristic assumption reminiscent of those used in subexponential classical factorization algorithms. It is currently not clear if the algorithm can lead to improved physical implementations in practice.

An Efficient Quantum Factoring Algorithm

TL;DR

This work proposes a quantum factoring scheme that reduces the per-run circuit size to gates and requires only independent executions, followed by polynomial-time classical post-processing. The method uses a multidimensional, Shor-like quantum procedure to sample near dual-lattice vectors, then employs lattice-reduction techniques (LLL) and an extended lattice construction to recover a short vector in , from which a nontrivial factor of the target is obtained via a gcd computation. The main contributions are the circuit-size improvement, the explicit lattice-based post-processing framework, and the conditional claims based on a number-theoretic heuristic; the practicality depends on whether the heuristic holds and on future optimizations, especially compared to optimized Shor implementations. The result suggests that under plausible assumptions, quantum resources for factoring could be reduced, although real-world performance remains to be demonstrated and verified against existing quantum factoring approaches.

Abstract

We show that -bit integers can be factorized by independently running a quantum circuit with gates for times, and then using polynomial-time classical post-processing. The correctness of the algorithm relies on a number-theoretic heuristic assumption reminiscent of those used in subexponential classical factorization algorithms. It is currently not clear if the algorithm can lead to improved physical implementations in practice.
Paper Structure (10 sections, 10 theorems, 34 equations)

This paper contains 10 sections, 10 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Statement of the result:
  3. Related work:
  4. Followup work:
  5. Acknowledgements:
  6. High level overview of the algorithm
  7. The quantum procedure
  8. Recovering a lattice from noisy samples of the dual
  9. Proof of the main theorem
  10. Fourier transform calculation

Key Result

Theorem 1.1

Let $N$ be an $n$-bit number and assume that for $d=\sqrt{n}$ and $O(\log n)$-bit numbers $b_1,\ldots,b_d$, there exists a vector in $\mathcal{L} \setminus \mathcal{L}_0$ of norm at most $T=\exp(O(\sqrt{n}))$. Then, there is a classical polynomial-time algorithm that outputs a non-trivial factor of

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 4.1: Pomerance01
  • Corollary 4.2
  • proof
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • proof
  • Corollary 4.5
  • Claim 5.1
  • ...and 14 more