Optimal compilation of parametrised quantum circuits

TL;DR

This paper investigates the optimal number of parameters in certain parametrised quantum circuits, demonstrating efficient solutions under specific restrictions and establishing the sufficiency of ZX-calculus rewrite rules for circuit equivalences.

Contribution

It proves the NP-hardness of general parameter minimization, identifies conditions for efficient optimization, and shows that existing ZX-calculus rules are sufficient for circuit equality proofs.

Findings

Efficient parameter minimization is possible for Clifford circuits with single-use parameters.

Simple 'fusion' rewrites are the only necessary transformations under well-behaved parameter conditions.

Existing optimization strategies are proven to be optimal within the specified circuit class.

Abstract

Parametrised quantum circuits contain phase gates whose phase is determined by a classical algorithm prior to running the circuit on a quantum device. Such circuits are used in variational algorithms like QAOA and VQE. In order for these algorithms to be as efficient as possible it is important that we use the fewest number of parameters. We show that, while the general problem of minimising the number of parameters is NP-hard, when we restrict to circuits that are Clifford apart from parametrised phase gates and where each parameter is used just once, we *can* efficiently find the optimal parameter count. We show that when parameter transformations are required to be sufficiently well-behaved, the only rewrites that reduce parameters correspond to simple 'fusions'. Using this we find that a previous circuit optimisation strategy by some of the authors [Kissinger, van de Wetering. PRA (2019)] finds the optimal number of parameters. Our proof uses the ZX-calculus. We also prove that the standard rewrite rules of the ZX-calculus suffice to prove any equality between parametrised Clifford circuits.