Graph-theoretic simplification of quantum circuits with the ZX-calculus

Tools

Duncan, Ross and Kissinger, Aleks and Perdrix, Simon and van de Wetering, John (2020) Graph-theoretic simplification of quantum circuits with the ZX-calculus. Quantum, 4. 279. ISSN 2521-327X (https://doi.org/10.22331/q-2020-06-04-279)

[thumbnail of Duncan-etal-Quantum-2020-Graph-theoretic-simplication-of-quantum-circuits-with-the-ZX-calculus]
Preview
 Text. Filename: Duncan_etal_Quantum_2020_Graph_theoretic_simplication_of_quantum_circuits_with_the_ZX_calculus.pdf
Final Published Version
License: Creative Commons Attribution 4.0 logo
Download (1MB)| Preview

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.

ORCID iDs

Duncan, Ross ORCID logoORCID: https://orcid.org/0000-0001-6758-1573, Kissinger, Aleks, Perdrix, Simon and van de Wetering, John;
CORE (COnnecting REpositories)