Approximate Block Diagonalization of Symmetric Matrices Using the D-Wave Advantage Quantum Annealer

Abstract

Approximate block diagonalization is a problem of transforming a given symmetric matrix as close to block diagonal as possible by symmetric permutations of its rows and columns. This problem arises as a preprocessing stage of various scientific calculations and has been shown to be NP-complete. In this paper, we consider solving this problem approximately using the D-Wave Advantage quantum annealer. For this purpose, several steps are needed. First, we have to reformulate the problem as a quadratic unconstrained binary optimization (QUBO) problem. Second, the QUBO has to be embedded into the physical qubit network of the quantum annealer. Third, and optionally, reverse annealing for improving the solution can be applied. We propose two QUBO formulations and four embedding strategies for the problem and discuss their advantages and disadvantages. Through numerical experiments, it is shown that the combination of domain-wall encoding and D-Wave's automatic embedding is the most efficient in terms of usage of physical qubits, while the combination of one-hot encoding and automatic embedding is superior in terms of the probability of obtaining a feasible solution. It is also shown that reverse annealing is effective in improving the solution for medium-sized problems.