A doubly stochastic matrices-based approach to optimal qubit routing

Swap mapping is a quantum compiler optimization that, by introducing \(\text {SWAP}\) gates, maps a logical quantum circuit to an equivalent physically implementable one. The physical implementability of a circuit is determined by the fulfillment of the hardware connectivity constraints. Therefore, the placement of the \(\text {SWAP}\) gates can be interpreted as a discrete optimization process. In this work, we employ a structure called doubly stochastic matrix, which is defined as a convex combination of permutation matrices. The intuition is that of making the decision process smooth. Doubly stochastic matrices are contained in the Birkhoff polytope, in which the vertices represent single permutation matrices. In essence, the algorithm uses smooth constrained optimization to slide along the edges of the polytope toward the potential solutions on the vertices. In the experiments, we show that the proposed algorithm, at the cost of additional computation time, can deliver significant depth reduction when compared to the state-of -the-art algorithm SABRE.