Dual-Matrix Domain-Wall: A Novel Technique for Generating Permutations by QUBO and Ising Models with Quadratic Sizes

TL;DR

This paper introduces the dual-matrix domain-wall encoding, a novel method for representing permutation problems in Ising and QUBO models that reduces quadratic terms and coefficient magnitudes, improving efficiency.

Contribution

The paper presents a new permutation encoding technique called dual-matrix domain-wall that significantly reduces quadratic terms and coefficients in Ising/QUBO models for permutation problems.

Findings

Reduces quadratic term count from $n^3-n^2$ to $6n^2-12n+4

Lowers maximum absolute coefficient from $2n-4$ to 2

Applicable to partial permutations and efficient permutation problem implementation

Abstract

The Ising model is defined by an objective function using a quadratic formula of qubit variables. The problem of an Ising model aims to determine the qubit values of the variables that minimize the objective function, and many optimization problems can be reduced to this problem. In this paper, we focus on optimization problems related to permutations, where the goal is to find the optimal permutation out of the $n!$ possible permutations of $n$ elements. To represent these problems as Ising models, a commonly employed approach is to use a kernel that utilizes one-hot encoding to find any one of the $n!$ permutations as the optimal solution. However, this kernel contains a large number of quadratic terms and high absolute coefficient values. The main contribution of this paper is the introduction of a novel permutation encoding technique called dual-matrix domain-wall, which significantly reduces the number of quadratic terms and the maximum absolute coefficient values in the kernel. Surprisingly, our dual-matrix domain-wall encoding reduces the quadratic term count and maximum absolute coefficient values from $n^3-n^2$ and $2n-4$ to $6n^2-12n+4$ and $2$, respectively. We also demonstrate the applicability of our encoding technique to partial permutations and Quadratic Unconstrained Binary Optimization (QUBO) models. Furthermore, we discuss a family of permutation problems that can be efficiently implemented using Ising/QUBO models with our dual-matrix domain-wall encoding.