Complete Equational Theories for the Sum-Over-Paths with Unbalanced Amplitudes
Matthew Amy · Electronic Proceedings in Theoretical Computer Science · 2023
Vilmart recently gave a complete equational theory for the balanced sum-over-paths over Toffoli-Hadamard circuits, and by extension Clifford+diag(1, ζ 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.