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A physical limitation in quantum circuit design is the fact that gates in aquantum system can only act on qubits that are physically adjacent in thearchitecture. To overcome this problem, SWAP gates need to be inserted to makethe circuit physically realizable. The nearest neighbour compliance problem(NNCP) asks for an optimal embedding of qubits in a given architecture suchthat the total number of SWAP gates to be inserted is minimized. In this paperwe study the NNCP on general quantum architectures. Building upon an existinglinear programming formulation, we show how the model can be reduced byexploiting the symmetries of the graph underlying the formulation. Theresulting model is equivalent to a generalized network flow problem and followsfrom an in-depth analysis of the automorphism group of specific Cayley graphs.As a byproduct of our approach, we show that the NNCP is polynomial timesolvable for several classes of symmetric quantum architectures. Numericaltests on various architectures indicate that the reduction in the number ofvariables and constraints is on average at least 90%. In particular, NNCPinstances on the star architecture can be solved for quantum circuits up to 100qubits and more than 1000 quantum gates within a very short computation time.These results are far beyond the computational capacity when solving theinstances without the exploitation of symmetries.

Exploiting Symmetries in Optimal Quantum Circuit Design