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.

翻译：Ising模型由基于量子比特变量的二次公式构成的目标函数定义。该模型的目标是确定最小化目标函数的变量量子比特值，许多优化问题均可归约为该问题。本文聚焦于与排列相关的优化问题，其目标是从$n$个元素的$n!$种可能排列中寻找最优排列。为将此类问题表示为Ising模型，常用方法采用基于独热编码的核函数，以在$n!$种排列中识别最优解。然而，该核函数包含大量二次项且系数绝对值较高。本文的主要贡献在于提出一种名为双矩阵畴壁的新型排列编码技术，该技术显著减少了核函数中的二次项数量与最大系数绝对值。令人惊讶的是，双矩阵畴壁编码将二次项数量和最大系数绝对值从$n^3-n^2$和$2n-4$分别降至$6n^2-12n+4$和$2$。此外，我们展示了该编码技术对部分排列及二次无约束二元优化（QUBO）模型的适用性。最后，我们讨论了可通过双矩阵畴壁编码的Ising/QUBO模型高效实现的一系列排列问题。