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A Complete Equational Theory for Quantum Circuits

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arxiv 2206.10577 v2 pith:A74JVDNC submitted 2022-06-21 quant-ph

classification quant-ph
keywords circuitscompletequantumequationalequationsgatesintroducetheory
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We introduce the first complete equational theory for quantum circuits. More precisely, we introduce a set of circuit equations that we prove to be sound and complete: two circuits represent the same unitary map if and only if they can be transformed one into the other using the equations. The proof is based on the properties of multi-controlled gates -- that are defined using elementary gates -- together with an encoding of quantum circuits into linear optical circuits, which have been proved to have a complete axiomatisation.

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  1. Completeness for flow-preserving rewrite rules

    quant-ph 2026-08 conditional novelty 7.0 of 10

    The flow-preserving rewrite rules of Figure 2 are complete for all MBQC-form ZX-diagrams with Pauli flow.

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