Improving solutions by embedding larger subproblems in a D-Wave quantum annealer

Abstract

Quantum annealing is a heuristic algorithm that solves combinatorial optimization problems, and D-Wave Systems Inc. has developed hardware implementation of this algorithm. However, in general, we cannot embed all the logical variables of a large-scale problem, since the number of available qubits is limited. In order to handle a large problem, qbsolv has been proposed as a method for partitioning the original large problem into subproblems that are embeddable in the D-Wave quantum annealer, and it then iteratively optimizes the subproblems using the quantum annealer. Multiple logical variables in the subproblem are simultaneously updated in this iterative solver, and using this approach we expect to obtain better solutions than can be obtained by conventional local search algorithms. Although embedding of large subproblems is essential for improving the accuracy of solutions in this scheme, the size of the subproblems are small in qbsolv since the subproblems are basically embedded by using an embedding of a complete graph even for sparse problem graphs. This means that the resource of the D-Wave quantum annealer is not exploited efficiently. In this paper, we propose a fast algorithm for embedding larger subproblems, and we show that better solutions are obtained efficiently by embedding larger subproblems.

Efficient partition of integer optimization problems with one-hot encoding

“…This partitioning method is similar to that developed in the literature 31 . While the extracted subproblems are embedded usnig complete-graph embedding in the literature 31 algorithm, which we developed in a previous study 45 . Details on how to achieve a multivalued partition using the subproblemembedding algorithm are explained in the Methods section.…”

“…The random partition does not adress whether an extracted subproblem contains feasible solutions for each S i or not. The subproblem-embedding algorithm proposed in the literature 45 is used for embedding a subproblem into the D-Wave quantum annealer. After optimizing the embedded subproblem using the D-Wave quantum annealer, the variables in the subproblem are replaced to the best solution among the 1, 000 solutions obtained using the quantum annealer.…”

“…We explain how to achieve the multivalued partition using the subproblem-embedding algorithm 45 . The key idea of the subproblem-embedding algorithm is to exclude binary variables that are not easily embedded from the subproblem and thereby reduce the computation time.…”

“…Because subproblems must be iteratively embedded, fast algorithms to embed larger subproblems are required for exploiting the potential of the quantum annealer. While it is reasonable to employ a completegraph embedding [42][43][44] for problems with dense interactions, the subproblem-embedding algorithm, which we developed in a previous study 45 , might be effective for improving solution accuracy of sparse problems.…”

Breaking limitation of quantum annealer in solving optimization problems under constraints

“…The connection between the physical qubits is sparse and limited on the chimera graph. Several embedding techniques are thus proposed, but the number of logical qubits, which represent the optimization problems to be solved, is drastically reduced 37 . In particular, the optimization problem, when it is written in terms of the Ising model, including fully or even partly connected spins, suffers from the smallness of the embeddable size on the chimera graph.…”