ZX-calculus for the working quantum computer scientist

TL;DR

The paper provides an accessible introduction to the ZX-calculus, a graphical language for quantum computing, and reviews recent advances, extensions, and theoretical foundations that enhance its applicability in quantum computation reasoning.

Contribution

It offers a comprehensive overview of ZX-calculus, including recent completeness results, extensions for mixed states and higher dimensions, and practical guidance for quantum computing applications.

Findings

Proves the Gottesman-Knill theorem graphically

Introduces extensions for Toffoli gates and mixed states

Shows ZX-calculus completeness for quantum reasoning

Abstract

The ZX-calculus is a graphical language for reasoning about quantum computation that has recently seen an increased usage in a variety of areas such as quantum circuit optimisation, surface codes and lattice surgery, measurement-based quantum computation, and quantum foundations. The first half of this review gives a gentle introduction to the ZX-calculus suitable for those familiar with the basics of quantum computing. The aim here is to make the reader comfortable enough with the ZX-calculus that they could use it in their daily work for small computations on quantum circuits and states. The latter sections give a condensed overview of the literature on the ZX-calculus. We discuss Clifford computation and graphically prove the Gottesman-Knill theorem, we discuss a recently introduced extension of the ZX-calculus that allows for convenient reasoning about Toffoli gates, and we discuss the recent completeness theorems for the ZX-calculus that show that, in principle, all reasoning about quantum computation can be done using ZX-diagrams. Additionally, we discuss the categorical and algebraic origins of the ZX-calculus and we discuss several extensions of the language which can represent mixed states, measurement, classical control and higher-dimensional qudits.