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Vilmart recently gave a complete equational theory for the balancedsum-over-paths over Toffoli-Hadamard circuits, and by extensionClifford+Rz(2pi/2^k) circuits. Their theory is based on the phase-freeZH-calculus which crucially omits the average rule of the full ZH-calculus,dis-allowing the local summation of amplitudes. Here we study the question ofcompleteness 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, variousformulations of the average and ortho rules of the ZH-calculus are sufficientto give complete equational theories over arbitrary rings and fields.

Complete Equational Theories for the Sum-Over-Paths with Unbalanced Amplitudes