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An Ising model is defined by a quadratic objective function known as the Hamiltonian, composed of spin variables that can take values of either $-1$ or $+1$. The goal is to assign spin values to these variables in a way that minimizes the value of the Hamiltonian. Ising models are instrumental in tackling many combinatorial optimization problems, leading to significant research in developing solvers for them. Notably, D-Wave Systems has pioneered the creation of quantum annealers, programmable solvers based on quantum mechanics, for these models. This paper introduces unit Ising models, where all non-zero coefficients of linear and quadratic terms are either $-1$ or $+1$. Due to the limited resolution of quantum annealers, unit Ising models are more suitable for quantum annealers to find optimal solutions. We propose a novel design methodology for unit Ising models to simulate logic circuits computing Boolean functions through integer linear programming. By optimizing these Ising models with quantum annealers, we can compute Boolean functions and their inverses. With a fixed unit Ising model for a logic circuit, we can potentially design Application-Specific Unit Quantum Annealers (ASUQAs) for computing the inverse function, which is analogous to Application-Specific Integrated Circuits (ASICs) in digital circuitry. For instance, if we apply this technique to a multiplication circuit, we can design an ASUQA for factorization of two numbers. Our findings suggest a powerful new method for compromising the RSA cryptosystem by leveraging ASUQAs in factorization., Comment: 16 pages, 9 figures, 2 tables

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., Comment: 26 pages, 9 figures