Global Synthesis of CNOT Circuits with Holes

TL;DR

This paper introduces a novel method for quantum circuit resynthesis called quantum combs, which allows for more flexible circuit transformations by 'cutting out' unresynthesizable gates, improving upon existing slicing techniques.

Contribution

The paper proposes quantum combs as an alternative to slicing for circuit resynthesis, extending the RowCol algorithm to handle quantum combs and demonstrating empirical improvements over naive methods.

Findings

Quantum combs enable direct resynthesis of circuits with holes.

Extended RowCol algorithm effectively handles quantum combs.

Empirical results show improved performance over slice-and-build methods.

Abstract

A common approach to quantum circuit transformation is to use the properties of a specific gate set to create an efficient representation of a given circuit's unitary, such as a parity matrix or stabiliser tableau, and then resynthesise an improved circuit, e.g. with fewer gates or respecting connectivity constraints. Since these methods rely on a restricted gate set, generalisation to arbitrary circuits usually involves slicing the circuit into pieces that can be resynthesised and working with these separately. The choices made about what gates should go into each slice can have a major effect on the performance of the resynthesis. In this paper we propose an alternative approach to generalising these resynthesis algorithms to general quantum circuits. Instead of cutting the circuit into slices, we "cut out" the gates we can't resynthesise leaving holes in our quantum circuit. The result is a second-order process called a quantum comb, which can be resynthesised directly. We apply this idea to the RowCol algorithm, which resynthesises CNOT circuits for topologically constrained hardware, explaining how we were able to extend it to work for quantum combs. We then compare the generalisation of RowCol using our method to the naive "slice and build" method empirically on a variety of circuit sizes and hardware topologies. Finally, we outline how quantum combs could be used to help generalise other resynthesis algorithms.