A doubly stochastic matrices-based approach to optimal qubit routing

TL;DR

This paper introduces a novel quantum circuit swap mapping method using doubly stochastic matrices to enable smooth optimization, resulting in significant depth reduction compared to existing algorithms.

Contribution

The work presents a new approach employing doubly stochastic matrices for optimal qubit routing, enhancing the efficiency of swap placement in quantum circuits.

Findings

Achieves significant depth reduction over SABRE.

Uses convex optimization within the Birkhoff polytope.

Trade-off between computation time and optimization quality.

Abstract

Swap mapping is a quantum compiler optimization that, by introducing 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 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.