# SZX-calculus: Scalable Graphical Quantum Reasoning

## TL;DR

The paper introduces SZX-calculus, an extension of ZX-calculus, enabling scalable, graphical reasoning about large quantum systems through register-based representations and matrix compactification.

## Contribution

It extends ZX-calculus to handle qubit registers and compact sub-diagrams, enhancing scalability for complex quantum computations.

## Key findings

- Proves soundness and completeness of SZX-calculus.
- Demonstrates applications in graph states.
- Shows utility in quantum error correction.

## Abstract

We introduce the Scalable ZX-calculus (SZX-calculus for short), a formal and compact graphical language for the design and verification of quantum computations. The SZX-calculus is an extension of the ZX-calculus, a powerful framework that captures graphically the fundamental properties of quantum mechanics through its complete set of rewrite rules. The ZX-calculus is, however, a low level language, with each wire representing a single qubit. This limits its ability to handle large and elaborate quantum evolutions. We extend the ZX-calculus to registers of qubits and allow compact representation of sub-diagrams via binary matrices. We show soundness and completeness of the SZX-calculus and provide two examples of applications, for graph states and error correcting codes.