Diagrammatic Analysis for Parameterized Quantum Circuits

TL;DR

This paper extends the ZX-calculus to better analyze parameterized quantum circuits, providing new rewrite rules and insights that facilitate understanding and designing quantum algorithms for applications like quantum chemistry and optimization.

Contribution

The work introduces formal rules for linear combinations of ZX-diagrams with complex coefficients, enhancing the analysis of parameterized quantum circuits beyond previous single-diagram methods.

Findings

Diagrammatic approach offers intuitive insights into algorithm structure.

New rewrite rules improve analysis of observable expectation values.

Potential to aid in designing more effective quantum circuit ansatze.

Abstract

Diagrammatic representations of quantum algorithms and circuits offer novel approaches to their design and analysis. In this work, we describe extensions of the ZX-calculus especially suitable for parameterized quantum circuits, in particular for computing observable expectation values as functions of or for fixed parameters, which are important algorithmic quantities in a variety of applications ranging from combinatorial optimization to quantum chemistry. We provide several new ZX-diagram rewrite rules and generalizations for this setting. In particular, we give formal rules for dealing with linear combinations of ZX-diagrams, where the relative complex-valued scale factors of each diagram must be kept track of, in contrast to most previously studied single-diagram realizations where these coefficients can be effectively ignored. This allows us to directly import a number useful relations from the operator analysis to ZX-calculus setting, including causal cone and quantum gate commutation rules. We demonstrate that the diagrammatic approach offers useful insights into algorithm structure and performance by considering several ansatze from the literature including realizations of hardware-efficient ansatze and QAOA. We find that by employing a diagrammatic representation, calculations across different ansatze can become more intuitive and potentially easier to approach systematically than by alternative means. Finally, we outline how diagrammatic approaches may aid in the design and study of new and more effective quantum circuit ansatze.