Completeness of Sum-Over-Paths for Toffoli-Hadamard and the Clifford Hierarchy

05/05/2022

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by   Renaud Vilmart, et al.

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The "Sum-Over-Paths" formalism is a way to symbolically manipulate linear maps that describe quantum systems, and is a tool that is used in formal verification. We give here a new set of rewrite rules for the formalism, and show that it is complete for "Toffoli-Hadamard", the simplest approximately universal fragment of quantum mechanics. We show that the rewriting is terminating, but not confluent (which is expected from the universality of the fragment). We do so using the connection between Sum-over-Paths and graphical language ZH-Calculus, and also show how the axiomatisation translates into the latter. Finally, we show how to enrich the rewrite system to reach completeness for the whole Clifford hierarchy.