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The "Sum-Over-Paths" formalism is a way to symbolically manipulate linearmaps that describe quantum systems, and is a tool that is used in formalverification of such systems. We give here a new set of rewrite rules for theformalism, and show that it is complete for "Toffoli-Hadamard", the simplestapproximately universal fragment of quantum mechanics. We show that therewriting is terminating, but not confluent (which is expected from theuniversality of the fragment). We do so using the connection betweenSum-over-Paths and graphical language ZH-Calculus, and also show how theaxiomatisation translates into the latter. Finally, we show how to enrich therewrite system to reach completeness for the dyadic fragments of quantumcomputation -- obtained by adding phase gates with dyadic multiples of $\pi$ tothe Toffoli-Hadamard gate-set -- used in particular in the Quantum FourierTransform.

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