  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.
