Using spectral graph theory to map qubits onto connectivity-limited devices

Abstract

In this thesis, we explore the use of spectral graph theory for mapping the logical qubits of quantum algorithms to the physical qubits of connectivity-limited devices. In particular, we propose an efficient approximate algorithm for making a circuit connectivity-compliant by adding a minimal number of connectivity-compliant SWAP gates. To this end, we generate a sequence of mappings such that the gates of the original algorithm can be placed, while being connectivity-compliant and adhering to the original ordering of the gates. We seek to find such a sequence while minimizing the total number of SWAP gates needed to transition from one mapping to the other. Taking inspiration from spectral graph drawing, we use an eigenvector of a graph Laplacian to place logical qubits at coordinate locations in one dimension. These placements are then mapped to physical qubits for a given connectivity. The graph from which the Laplacian is taken from is designed with higher edge weights between pairs of qubits that should be placed close together. The specific way in which these weights are chosen depends on a variety of factors relating to the architecture and circuit. The proposal and evaluation of novel edge weight calculations, along with the application of spectral drawing methods, is the main contribution of this thesis. Focusing mainly on the relatively restrictive one-dimensional, linear nearest neighbor architecture, our results provide relevancy to near term, intermediate scale devices.