Niel De Beaudrap (University of Oxford), Ross Duncan (University of Strathclyde/Cambridge Quantum Computing), Dominic Horsman (Université Grenoble Alpes), Simon Perdrix (CNRS, LORIA, Université de Lorraine)
The ZX calculus is an abstract mathematical tool to represent — and importantly, to “reason” (or calculate) with — tensors of a sort that frequently occur in quantum computational theory. From the outset, the ZX calculus was developed to be universal, and more recently, it has been extended to be made complete. These features make the ZX calculus attractive as a potential component of an optimising compiler for quantum computers — provided that one can determine whether and how a transformation denoted by a ZX term can be deterministically realised. Until recently, it was not known how to do so, unless the ZX term was in a certain sense “circuit-like”. We present an abstract model of quantum computation, the Pauli Fusion model, whose primitive operations correspond closely to generators of the ZX calculus (and are also straightforward abstractions of the basic operations in some leading proposed quantum technologies). These operations have non-deterministic heralded effects, similarly to measurement-based quantum computation. We describe sufficient conditions for Pauli Fusion procedures to be deterministically realisable, so that the quantum transformation performed is independent of the non-deterministic outcomes. This allows us to give an operational model for the realisation of ZX terms beyond the circuit model.