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

TL;DR

This paper develops complete equational theories for unbalanced sum-over-paths in quantum circuits, enabling local amplitude summation and broad applicability across rings and fields.

Contribution

It introduces a concrete syntax for unbalanced sum-over-paths and demonstrates that interference, average, and ortho rules suffice for completeness.

Findings

Complete equational theories for unbalanced sum-over-paths established.

Theories support local amplitude summation in quantum circuit analysis.

Applicability extends to arbitrary rings and fields.

Abstract

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.