Embedding and Weight Distribution for Quantum Annealing

Abstract

Before being able to calculate on the D-Wave machine, its very restricted structure requires the embedding of the original problem graph onto the Chimera hardware graph. A precalculated embedding of a complete graph enables to map all problems with the same number of nodes or less straightforwardly. The problem of finding the largest complete graph minor and its embedding scheme in a Chimera graph with broken qubits can be formulated as an optimization problem, more precisely as a matching problem with additional linear constraints. Although being NP-hard in general it is fixed parameter tractable in the number of broken qubits. By dropping specific matches the problem can be simplified. Some preliminary results comparing this heuristic approach to exact optimization are shown. After the structural embedding the actual embedded Ising model needs to be constructed from the original problem coefficient values, such that the minima of both are equivalent. That means in the solution of the embedded Ising model the values for each single qubit embedding should be synchronized. The resulting constraints can be derived to a graph property related to expansion, which is efficient to solve in the embedding framework. First results show an improvement over standard methods with respect to coefficient ratio

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