Variational quantum solutions to the Shortest Vector Problem

Martin R. Albrecht, Miloš Prokop, Yixin Shen, and Petros Wallden

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 ≈ 10^3 noisy qubits such instances can be tackled.

Available format(s)
Publication info
Preprint.
Keywords
Shortest Vector ProblemVariational Quantum Algorithms
Contact author(s)
martin albrecht @ rhul ac uk
m prokop @ sms ed ac uk
yixin shen @ rhul ac uk
petros wallden @ ed ac uk
History
2022-05-20: revised
See all versions
Short URL
https://ia.cr/2022/233

CC BY

BibTeX

@misc{cryptoeprint:2022/233,
author = {Martin R.  Albrecht and Miloš Prokop and Yixin Shen and Petros Wallden},
title = {Variational quantum solutions to the Shortest Vector Problem},
howpublished = {Cryptology ePrint Archive, Paper 2022/233},
year = {2022},
note = {\url{https://eprint.iacr.org/2022/233}},
url = {https://eprint.iacr.org/2022/233}
}

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