Variational quantum solutions to the Shortest Vector Problem

TL;DR

This paper investigates using NISQ quantum devices to solve the Shortest Vector Problem in lattices, establishing new bounds and methods to handle larger instances than previously possible, with implications for post-quantum cryptography.

Contribution

It introduces novel bounds and mappings for quantum algorithms to solve SVP, enabling larger lattice dimensions to be tackled on NISQ devices.

Findings

Solved SVP in dimension up to 28 in quantum emulation

Proposed new bounds for qubits needed per lattice dimension

Estimated that ~1000 noisy qubits could handle hard lattice instances

Abstract

A fundamental computational problem is to find a shortest non-zero vector in Euclidean lattices, a problem known as the Shortest Vector Problem (SVP). This problem is believed to be hard even on quantum computers and thus plays a pivotal role in post-quantum cryptography. In this work we explore how (efficiently) Noisy Intermediate Scale Quantum (NISQ) devices may be used to solve SVP. Specifically, we map the problem to that of finding the ground state of a suitable Hamiltonian. In particular, (i) we establish new bounds for lattice enumeration, this allows us to obtain new bounds (resp.~estimates) for the number of qubits required per dimension for any lattices (resp.~random q-ary lattices) to solve SVP; (ii) we exclude the zero vector from the optimization space by proposing (a) a different classical optimisation loop or alternatively (b) a new mapping to the Hamiltonian. These improvements allow us to solve SVP in dimension up to 28 in a quantum emulation, significantly more than what was previously achieved, even for special cases. Finally, we extrapolate the size of NISQ devices that is required to be able to solve instances of lattices that are hard even for the best classical algorithms and find that with approximately $10^3$ noisy qubits such instances can be tackled.