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.
Highlights
Combinatorial optimization problems, the minimization of cost functions with discrete variables, have significant real-world applications
In order to obtain better solutions, it is essential to search for as many local minima as possible
We showed that better solutions are obtained efficiently by embedding larger subproblems for the spin-glass and ferromagnetic models on the cubic lattice with 10 × 10 × 10 logical variables
Summary
Combinatorial optimization problems, the minimization of cost functions with discrete variables, have significant real-world applications. The cost function of a combinatorial optimization problem can generally be mapped to the Hamiltonian of a classical Ising model[1]. Where H 0 is the classical Hamiltonian which represents the cost function to be minimized, and Hq is the quantum fluctuation term for which the ground state is trivial. The system will remain close to the instantaneous ground state of the time-dependent Hamiltonian if the system changes sufficiently slowly and if the adiabatic condition13,. By setting the annealing time τ large enough, we obtain the ground state of the classical Hamiltonian H 0, which represents the optimal solution. The current version of D-Wave quantum annealer (D-Wave 2000Q) implements QA with a transverse magnetic field: Nq
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