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 applies 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 the 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 n3−n2 and 2n−4 to 6n2−12n+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.
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 applies 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 the 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 n3−n2 and 2n−4 to 6n2−12n+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.
To tackle combinatorial optimization problems using an Ising machine, the objective function and constraints must be mapped onto a quadratic unconstrained binary optimization (QUBO) model. While QUBO involves binary variables, combinatorial optimization problems frequently include integer variables, which require encoding by binary variables. This process, known as binary-integer encoding, includes various methods, one of which is domain-wall encoding - a recently proposed approach. Experiments on a quantum annealing machine have demonstrated that domain-wall encoding outperforms the commonly used one-hot encoding in terms of objective function value and the probability of obtaining the optimal solution. In a digital Ising machine, domain-wall encoding required less computation time to reach optimal solutions compared to one-hot encoding. However, its practical effectiveness in digital Ising machines remains unclear. To address this uncertainty, the performance of one-hot and domain-wall encoding methods was evaluated on a digital Ising machine using the quadratic knapsack problem (QKP). The comparison focused on the dependency of penalty coefficient and sensitivity to computation time. Domain-wall encoding demonstrated a higher feasible solution rate when relative penalty coefficients for the two constraint terms were adjusted, a strategy not commonly used in previous studies. Additionally, domain-wall encoding obtained higher performance practical evaluation metrics for QKPs with large knapsack capacities compared to one-hot encoding. Furthermore, it was observed to be more sensitive to computation time than one-hot encoding.
Many combinatorial optimization problems can be formulated as a problem to determine the order of sequence or to find a corresponding mapping of the objects. We call such problems permutation-based optimization problems. Many such problems can be formulated as a quadratic unconstrained binary optimization (QUBO) or Ising model by introducing a penalty coefficient to the permutation constraint terms. While classical and quantum annealing approaches have been proposed to solve QUBOs to date, they face issues with optimality and feasibility. Here we treat a given QUBO solver as a black box and propose techniques to enhance its performance. First, to ensure an effective search for good quality solutions, a smooth energy landscape is needed; we propose a data scaling approach that reduces the amplitudes of the input without compromising optimality. Second, we need to tune the penalty coefficient. In this paper, we illustrate that for certain problems, it suffices to tune the parameter by performing random sampling on the penalty coefficients to achieve good performance. Finally, to handle possible infeasibility of the solution, we introduce a polynomial-time projection algorithm. We apply these techniques along with a divide-and-conquer strategy to solve some large-scale permutation-based problems and present results for TSP and QAP.
ABSTRACT An Ising model is a mathematical model defined by an objective function comprising a quadratic formula of multiple spin variables, each taking values of either or . The task of determining a spin value assignment to these variables that minimizes the resulting value of an Ising model is a challenging optimization problem. Recently, quantum annealers, consisting of qubit cells interconnected according to principles of quantum mechanics, have emerged as a solution for tackling such problems. Ising models characterized by fewer quadratic terms are preferable as they reduce the resource requirements of quantum annealers. Additionally, it is advantageous for the absolute values of coefficients associated with linear and quadratic terms to be small to facilitate the discovery of good solutions, given the inherent limitations in the resolution of quantum annealers. The primary contribution of this article lies in presenting Ising models tailored for solving the ‐Queens puzzle. The conventional Ising model for this puzzle involves quadratic terms, with the maximum absolute value of coefficients being . Our novel Ising model significantly reduces the number of quadratic terms to only , with a maximum absolute coefficient of 6. Furthermore, we provide embedding results for a quantum annealer D‐Wave Advantage utilizing a Pegasus graph . We succeeded in embedding our novel Ising model for up to the 21‐Queens puzzle, while the conventional Ising model can be embedded only for up to the 14‐Queens puzzle.
QUBO is the input language of quantum computers based on quantum annealing such as the D-Wave systems and of the "quantum-inspired" dedicated hardware such as Fujitsu’s Digital Annealing Unit. We propose a new way to represent integer variables in QUBO together with constraints on those variables by using unary/domain-wall encoding in order to solve constrained optimization problem by quantum annealing. We detail in particular how to encode permutation constraints on integers within the unary/domain-wall encoding scheme, as these constraints are omnipresent in complex combinatorial problems. This new encoding in QUBO is interesting in the context of quantum annealing because it minimizes interactions between variables, and therefore connections needed between qubits in hardware. When compared to the usual one-hot encoding of integers in QUBO, we show than the new encoding scheme divides by nearly a factor 2 the number of connections between qubits for complex problems involving permutation constraints and makes it possible to cope with larger problem instances on current quantum and quantum-inspired hardware.
Quadratic Unconstrained Binary Optimization (QUBO) is a standard NP-hard optimization problem. Recently, it has gained renewed interest through quantum computing, as QUBOs directly reduce to the Ising model, on which quantum annealing devices are based. We introduce a QUBO formulation for permutations using compare-exchange networks, with only O(nlog2n) binary variables. This is a substantial improvement over the standard permutation matrix encoding, which requires n2 variables and has a much denser interaction graph. A central feature of our approach is uniformity: each permutation corresponds to a unique variable assignment, enabling unbiased sampling.
Our construction also allows additional constraints, including fixed points and parity. Moreover, it provides a representation of permutations that supports the operations multiplication and inversion, and also makes it possible to check the order of a permutation. This can be used to uniformly generate permutations of a given order or, for example, permutations that commute with a specified permutation. To our knowledge, this is the first result linking oblivious compare-exchange networks with QUBO encodings. While similar functionality can be achieved using permutation matrices, our method yields QUBOs that are both smaller and sparser. We expect this method to be practically useful in areas where unbiased sampling of constrained permutations is important, including cryptography and combinatorial design.
A quadratic unconstrained binary optimization (QUBO) model is defined by an objective function that consists of a quadratic formula involving multiple binary variables. The goal of the QUBO problem is to find an optimal assignment of values to the variables that minimizes the objective function. One of the most well-known problems that can be reduced to a QUBO problem is the N-Queens problem. The N-Queens problem seeks to find a configuration of n queens on an n×n chessboard such that no two queens can attack each other. It is widely recognized that a conventional technique can be used to design a QUBO model with an n×n matrix of binary variables representing the chessboard, where a queen is placed in a particular position if and only if the corresponding binary variable is set to 1. This technique involves incorporating terms in the objective function to ensure that the sum of each row and column is 1, and the sum of each diagonal and anti-diagonal is either 0 or 1. The resulting reduced QUBO model contains 35n3−2n2+31n quadratic terms. Due to the simplicity and straightforwardness of designing such a QUBO model, there are numerous resources available that explain this model as an easy example of QUBO models. However, to the best of our knowledge, no fundamentally different approach to designing a QUBO model for the N-Queens problem has been discovered. The prevailing view within the community is that this approach is the only method for designing a QUBO model for the N-Queens problem, and no superior alternative has been found. In this paper, we will present a counterintuitive result that demonstrates a more efficient QUBO model for the N-Queens problem, requiring fewer quadratic terms. Surprisingly, this alternative model involves only 12n2−24n+12 quadratic terms, which is smaller than the conventional QUBO model whenever n is greater than or equal to 7. This discovery marks a significant breakthrough in the theoretical challenge of minimizing a QUBO model for solving the N-Queens problem.