Cryptology ePrint Archive: Report 2022/233

Variational quantum solutions to the Shortest Vector Problem

Martin R. Albrecht and Miloš Prokop and 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.

Category / Keywords: Shortest Vector Problem, Variational Quantum Algorithms

Date: received 23 Feb 2022, last revised 20 May 2022

Contact author: martin albrecht at rhul ac uk, m prokop at sms ed ac uk, yixin shen at rhul ac uk, petros wallden at ed ac uk

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Version: 20220520:125434 (All versions of this report)

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