Space-Optimized and Experimental Implementations of Regev's Quantum Factoring Algorithm

TL;DR

This paper presents space-efficient quantum circuit implementations of Regev's lattice-based factoring algorithm, demonstrating practical feasibility through simulations and experiments on superconducting quantum hardware.

Contribution

It introduces a qubit reuse method that significantly reduces space complexity and provides the first experimental implementation of Regev's quantum factoring algorithm.

Findings

Reduced space complexity from O(n^{3/2}) to O(n log n)

Successfully factored N=35 on a superconducting quantum computer

Demonstrated practical resource scaling and time-space trade-offs

Abstract

The integer factorization problem (IFP) underpins the security of RSA, yet becomes efficiently solvable on a quantum computer through Shor's algorithm. Regev's recent high-dimensional variant reduces the circuit size through lattice-based post-processing, but introduces substantial space overhead and lacks practical implementations. Here, we propose a qubit reuse method by intermediate-uncomputation that significantly reduces the space complexity of Regev's algorithm, inspired by reversible computing. Our basic strategy lowers the cost from \( O(n^{3/2}) \) to \( O(n^{5/4}) \), and refined strategies achieve \( O(n \log n) \)which is a space lower bound within this model. Simulations demonstrate the resulting time-space trade-offs and resource scaling. Moreover, we construct and compile quantum circuits that factor \( N = 35 \), verifying the effectiveness of our method through noisy simulations. A more simplified experimental circuit for Regev's algorithm is executed on a superconducting quantum computer, with lattice-based post-processing successfully retrieving the factors. These results advance the practical feasibility of Regev-style quantum factoring and provide guidance for future theoretical and experimental developments.