  Vilmart recently gave a complete equational theory for the balanced
sum-over-paths over Toffoli-Hadamard circuits, and by extension
Clifford+Rz(2pi/2^k) circuits. Their theory is based on the phase-free
ZH-calculus which crucially omits the average rule of the full ZH-calculus,
dis-allowing the local summation of amplitudes. Here we study the question of
completeness in unbalanced path sums which naturally support local summation.
We give a concrete syntax for the unbalanced sum-over-paths and show that,
together with symbolic multilinear algebra and the interference rule, various
formulations of the average and ortho rules of the ZH-calculus are sufficient
to give complete equational theories over arbitrary rings and fields.
