---
# System prepended metadata

title: '[WIP][Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus](https://arxiv.org/abs/1902.03178)'
tags: [' quantum2019', ' quantum computing', ' 2019', ' local complementation', quantum circuit optimization, ' zx-calculus']

---

---
tags: quantum circuit optimization, local complementation, zx-calculus, quantum computing, quantum2019, 2019
---

# [WIP][Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus](https://arxiv.org/abs/1902.03178)
Ross Duncan, Aleks Kissinger, Simon Perdrix, John van de Wetering

## 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.


## Backgrounds and Contributions


## Methods


## Open Problems