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

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

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)
2 Downloads (Pure)


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.
Original languageEnglish
Article number279
Number of pages33
Publication statusPublished - 27 May 2020


  • quantum science
  • ZX-calculus
  • quantum circuits
  • ZX-diagrams
  • graph


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