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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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:

For a 2-minute intro to PyZX, see this YouTube video:

If that leaves you wanting more, there is also a 40-minute talk about this and more on YouTube:

► BibTeX data

► References

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