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

Ross Duncan1,2, 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

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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.

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

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Cited by

[1] Aleks Kissinger and John van de Wetering, "Reducing T-count with the ZX-calculus", arXiv:1903.10477.

[2] Aleks Kissinger and Arianne Meijer-van de Griend, "CNOT circuit extraction for topologically-constrained quantum memories", arXiv:1904.00633.

[3] Raban Iten, Oliver Reardon-Smith, Luca Mondada, Ethan Redmond, Ravjot Singh Kohli, and Roger Colbeck, "Introduction to UniversalQCompiler", arXiv:1904.01072.

[4] Niel de Beaudrap and Dominic Horsman, "The ZX calculus is a language for surface code lattice surgery", arXiv:1704.08670.

[5] Alexander Cowtan, Silas Dilkes, Ross Duncan, Will Simmons, and Seyon Sivarajah, "Phase Gadget Synthesis for Shallow Circuits", arXiv:1906.01734.

[6] Aleks Kissinger and John van de Wetering, "PyZX: Large Scale Automated Diagrammatic Reasoning", arXiv:1904.04735.

[7] John van de Wetering and Sal Wolffs, "Completeness of the Phase-free ZH-calculus", arXiv:1904.07545.

[8] Titouan Carette, Dominic Horsman, and Simon Perdrix, "SZX-calculus: Scalable Graphical Quantum Reasoning", arXiv:1905.00041.

[9] Titouan Carette, Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart, "Completeness of Graphical Languages for Mixed States Quantum Mechanics", arXiv:1902.07143.

[10] Cole Comfort, "Circuit Relations for Real Stabilizers: Towards TOF+H", arXiv:1904.10614.

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[12] Stach Kuijpers, John van de Wetering, and Aleks Kissinger, "Graphical Fourier Theory and the Cost of Quantum Addition", arXiv:1904.07551.

[13] Aleks Kissinger and John van de Wetering, "Universal MBQC with generalised parity-phase interactions and Pauli measurements", arXiv:1704.06504.

[14] Niel de Beaudrap, Ross Duncan, Dominic Horsman, and Simon Perdrix, "Pauli Fusion: a Computational Model to Realise Quantum Transformations from ZX Terms", arXiv:1904.12817.

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[18] Hector Miller-Bakewell, "Entanglement and Quaternions: The graphical calculus ZQ", arXiv:2003.09999.

[19] Miriam Backens, Hector Miller-Bakewell, Giovanni de Felice, Leo Lobski, and John van de Wetering, "There and back again: A circuit extraction tale", arXiv:2003.01664.

[20] Niel de Beaudrap, Xiaoning Bian, and Quanlong Wang, "Fast and effective techniques for T-count reduction via spider nest identities", arXiv:2004.05164.

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