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

Ross Duncan; Aleks Kissinger; Simon Perdrix; John van de Wetering

doi:10.22331/q-2020-06-04-279

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

Ross Duncan^1,2, Aleks Kissinger³, Simon Perdrix⁴, and John van de Wetering⁵

¹University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, UK
²Cambridge Quantum Computing Ltd, 9a Bridge Street, Cambridge CB2 1UB, UK
³Department of Computer Science, University of Oxford
⁴CNRS LORIA, Inria-MOCQUA, Université de Lorraine, F 54000 Nancy, France
⁵Institute for Computing and Information Sciences, Radboud University Nijmegen

Published:	2020-06-04, volume 4, page 279
Eprint:	arXiv:1902.03178v6
Doi:	https://doi.org/10.22331/q-2020-06-04-279
Citation:	Quantum 4, 279 (2020).

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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.

Popular summary

Quantum circuits are a de facto assembly language for quantum software. Programs are described as list of primitive operations, or gates, which are run in sequence on a quantum computer to perform a computation. Just like with classical software, there is more that one way to write a program to do the same job, and so it's important to find programs that do that job as quickly and cheaply as possible. Looking at quantum circuits just as lists of gates doesn't tell us a whole lot about what computation is being performed, or how it might be optimised. However, if we "break open" quantum gates, we see a rich graphical/algebraic structure inside called the ZX-calculus.

In this paper, we give a technique for optimising quantum circuits that first breaks the gates open to reveal a graph-like structure underneath. We then give a strategy for reducing these graphs in size without changing the computation they represent, then extracting a new, smaller circuit out of the result.

We have implemented the technique described in this paper in a Python tool called PyZX:
https://github.com/Quantomatic/pyzx

For a 2-minute intro to PyZX, see this YouTube video: https://www.youtube.com/watch?v=iC-KVdB8pf0

If that leaves you wanting more, there is also a 40-minute talk about this and more on YouTube: https://www.youtube.com/watch?v=JafI_LZts2g

► BibTeX data

@article{Duncan2020graphtheoretic,
  doi = {10.22331/q-2020-06-04-279},
  url = {https://doi.org/10.22331/q-2020-06-04-279},
  title = {Graph-theoretic {S}implification of {Q}uantum {C}ircuits with the {ZX}-calculus},
  author = {Duncan, Ross and Kissinger, Aleks and Perdrix, Simon and van de Wetering, John},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {4},
  pages = {279},
  month = jun,
  year = {2020}
}

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