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

Author:

Duncan Ross12ORCID,Kissinger Aleks3ORCID,Perdrix Simon4ORCID,van de Wetering John5ORCID

Affiliation:

1. University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, UK

2. Cambridge Quantum Computing Ltd, 9a Bridge Street, Cambridge CB2 1UB, UK

3. Department of Computer Science, University of Oxford

4. CNRS LORIA, Inria-MOCQUA, Université de Lorraine, F 54000 Nancy, France

5. Institute for Computing and Information Sciences, Radboud University Nijmegen

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.

Publisher

Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften

Subject

Physics and Astronomy (miscellaneous),Atomic and Molecular Physics, and Optics

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