Complete Equational Theories for the Sum-Over-Paths with Unbalanced Amplitudes
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Abstract
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.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.000 |
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| Science and technology studies | 0.001 | 0.002 |
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| Open science | 0.003 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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