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Journal Articles Quantum Year : 2020

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

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Aleks Kissinger
• Function : Author
Simon Perdrix

#### 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.

### Dates and versions

hal-02995364 , version 1 (09-11-2020)

### Identifiers

• HAL Id : hal-02995364 , version 1
• ARXIV :
• DOI :

### Cite

Ross Duncan, Aleks Kissinger, Simon Perdrix, John van de Wetering. Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus. Quantum, 2020, 4, pp.279. ⟨10.22331/q-2020-06-04-279⟩. ⟨hal-02995364⟩

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