### 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 naïve `cut-and-resynthesise' methods.

Original language | English |
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Article number | 279 |

Number of pages | 33 |

Journal | Quantum |

Volume | 4 |

DOIs | |

Publication status | Published - 4 Jun 2020 |

### Keywords

- quantum science
- ZX-calculus
- quantum circuits
- ZX-diagrams
- graph

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## Profiles

## Ross Duncan

- Computer And Information Sciences - Research Fellow
- Scottish Informatics and Computer Science Alliance

Person: Academic, Research Only

## Cite this

Duncan, R., Kissinger, A., Perdrix, S., & van de Wetering, J. (2020). Graph-theoretic simplification of quantum circuits with the ZX-calculus.

*Quantum*,*4*, [279]. https://doi.org/10.22331/q-2020-06-04-279