We introduce a method for solving a quadratic unconstrained binary optimization (QUBO) with the two-way one-hot constraints by dividing the QUBO into parts and solving it with an Ising machine. The one-hot constraint is a constraint condition in that only one binary variable takes the value one for a set of multiple binary variables. The two-way one-hot constraint imposes one-hot constraint conditions on every row and every column of a two-dimensional array of binary variables with equal numbers of rows and columns. For example, traveling salesman problems (TSPs) have two-way one-hot constraints. We propose two methods to solve a TSP by dividing the cities into clusters based on information in the QUBO matrix. The proposed methods decompose cities into clusters and solve TSPs for each cluster. We solve TSPs such that the cities are distributed in clusters and compare the results of the two proposed methods with the results of solving the problem without partitioning. The results show that the proposed methods are robust to the coefficient parameters in the Hamiltonian of QUBO and can obtain a solution closer to the optimal solution when solving the problem without partitioning is hard. We also discuss the application of the methods proposed in the TSPs to the quadratic assignment problems and to the problems of ordering things.
We consider the problem of transforming a given symmetric matrix into a nearly block diagonal form by permutation of its rows and columns. Such a transformation is useful as preconditioning to accelerate the convergence of an eigenvalue solver, but the problem of finding an optimal permutation that maximizes the Frobenius norms of the diagonal blocks is NP-complete. We formulate this problem as QUBO (Quadratic Unconstrained Binary Optimization) and solve it using D-Wave Advantage quantum annealing machine. Experimental results on small problems show that the true minimum can be obtained with high probability. We also discuss how to improve the mapping of the problem onto the physical qubit network to increase the size of the problems that can be solved.