Qubo model for the Closest Vector Problem

TL;DR

This paper demonstrates a polynomial-time reduction of the closest vector problem (CVP) for lattices to a quadratic unconstrained binary optimization (QUBO) problem, enabling potential quantum or classical optimization approaches.

Contribution

The paper introduces a novel polynomial-time reduction from CVP to QUBO, detailing the number of binary variables and coefficient sizes involved.

Findings

CVP reduces to QUBO in polynomial time

Number of binary variables is O(n^2(log(n)+log(b)))

Coefficient size is O(n(log(n)+log(b)))

Abstract

In this paper we consider the closest vector problem (CVP) for lattices $\Lambda \subseteq \mathbb{Z}^n$ given by a generator matrix $A\in \mathcal{M}_{n\times n}(\mathbb{Z})$. Let $b>0$ be the maximum of the absolute values of the entries of the matrix $A$. We prove that the CVP can be reduced in polynomial time to a quadratic unconstrained binary optimization (QUBO) problem in $O(n^2(\log(n)+\log(b)))$ binary variables, where the length of the coefficients in the corresponding quadratic form is $O(n(\log(n)+\log(b)))$.