# Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus

## TL;DR

This paper introduces a novel quantum circuit optimization method using ZX-calculus, simplifying diagrams through graph transformations and enabling efficient circuit extraction with improved size and depth bounds.

## Contribution

The paper develops a new ZX-calculus-based approach for quantum circuit simplification, including a graph-theoretic extraction method and improved bounds for Clifford and Clifford+T circuits.

## Key findings

- Simplifies quantum circuits via ZX-diagram transformations.
- Provides a deterministic circuit extraction method from simplified ZX-diagrams.
- Achieves smaller circuit size and depth bounds for Clifford and Clifford+T circuits.

## Abstract

We present a completely new approach to quantum circuit optimisation, based on the ZX-calculus. We first interpret quantum circuits as ZX-diagrams, which provide a flexible, lower-level language for describing quantum computations graphically. Then, using the rules of the ZX-calculus, we give a simplification strategy for ZX-diagrams based on the two graph transformations of local complementation and pivoting and show that the resulting reduced diagram can be transformed back into a quantum circuit. While little is known about extracting circuits from arbitrary ZX-diagrams, we show that the underlying graph of our simplified ZX-diagram always has a graph-theoretic property called generalised flow, which in turn yields a deterministic circuit extraction procedure. For Clifford circuits, this extraction procedure yields a new normal form that is both asymptotically optimal in size and gives a new, smaller upper bound on gate depth for nearest-neighbour architectures. For Clifford+T and more general circuits, our technique enables us to to `see around' gates that obstruct the Clifford structure and produce smaller circuits than naive 'cut-and-resynthesise' methods.