Equivalence Checking of Quantum Circuits with the ZX-Calculus

TL;DR

This paper evaluates the ZX-calculus as a tool for quantum circuit equivalence checking, demonstrating its capabilities, limitations, and comparing it with other methods through a detailed case study.

Contribution

It expands the ZX-calculus approach for equivalence checking, assesses its effectiveness, and compares it with existing methods within a practical tool.

Findings

ZX-calculus can verify quantum circuit equivalences but is not complete for circuits with ancillary qubits.

The method is effective for verifying compilation and optimization results.

Compared with path-sums and decision diagrams, ZX-calculus offers a complementary approach.

Abstract

As state-of-the-art quantum computers are capable of running increasingly complex algorithms, the need for automated methods to design and test potential applications rises. Equivalence checking of quantum circuits is an important, yet hardly automated, task in the development of the quantum software stack. Recently, new methods have been proposed that tackle this problem from widely different perspectives. One of them is based on the ZX-calculus, a graphical rewriting system for quantum computing. However, the power and capability of this equivalence checking method has barely been explored. The aim of this work is to evaluate the ZX-calculus as a tool for equivalence checking of quantum circuits. To this end, it is demonstrated how the ZX-calculus based approach for equivalence checking can be expanded in order to verify the results of compilation flows and optimizations on quantum circuits. It is also shown that the ZX-calculus based method is not complete$\unicode{x2014}$especially for quantum circuits with ancillary qubits. In order to properly evaluate the proposed method, we conduct a detailed case study by comparing it to two other state-of-the-art methods for equivalence checking: one based on path-sums and another based on decision diagrams. The proposed methods have been integrated into the publicly available QCEC tool (https://github.com/cda-tum/qcec) which is part of the Munich Quantum Toolkit (MQT).