An effcient variational quantum Korkin-Zolotarev algorithm for solving shortest vector problems

TL;DR

This paper introduces a variational quantum algorithm that efficiently solves the shortest vector problem in lattice cryptography, enabling larger problem instances to be tackled with near-term quantum devices.

Contribution

The paper proposes the VQKZ algorithm that reduces qubit requirements and transforms SVP into manageable subproblems, improving scalability and performance over existing quantum methods.

Findings

Solves SVP instances with 61.39% larger lattices than previous methods

Outperforms existing algorithms in solution vector length

Enables near-term quantum devices to handle larger cryptographic problems

Abstract

Noisy intermediate-scale quantum cryptanalysis focuses on the capability of near-term quantum devices to solve the mathematical problems underlying cryptography, and serves as a cornerstone for the design of post-quantum cryptographic algorithms. For the shortest vector problem (SVP), which is one of the computationally hard problems in lattice-based cryptography, existing near-term quantum cryptanalysis algorithms map the problem onto a fully-connected quantum Ising Hamiltonian, and obtain the solution by optimizing for the first excited state. However, as the quantum system scales with the problem size, determining the first excited state becomes intractable due to the exponentially increased complexity for large-scale SVP instances. In this paper, we propose a variational quantum Korkin-Zolotarev (VQKZ) algorithm, which significantly reduces the qubit requirement for solving the SVP. Specifically, by transforming the original SVP into a series of subproblems on projected sublattices, the proposed VQKZ algorithm enables near-term quantum devices to solve SVP instances with lattice dimensions 61.39% larger than those solvable by previous methods. Furthermore, numerical simulations demonstrate that the proposed VQKZ algorithm can significantly outperform existing methods in terms of the length of solution vectors.