Completeness of Sum-Over-Paths for Toffoli-Hadamard and the Dyadic Fragments of Quantum Computation

TL;DR

This paper develops a complete rewrite system for the Sum-Over-Paths formalism in quantum computing, specifically for the Toffoli-Hadamard fragment and dyadic phases, aiding formal verification of quantum systems.

Contribution

It introduces a new set of rewrite rules that are complete for the Toffoli-Hadamard fragment and extends to dyadic phases, connecting Sum-Over-Paths with ZH-Calculus.

Findings

Rewritings are terminating but not confluent.

The formalism is complete for Toffoli-Hadamard fragment.

Extension to dyadic phases achieves broader completeness.

Abstract

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 of such systems. 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 dyadic fragments of quantum computation -- obtained by adding phase gates with dyadic multiples of $\pi$ to the Toffoli-Hadamard gate-set -- used in particular in the Quantum Fourier Transform.