ZX-Flow: A Flexible Criterion for Deterministic Computation with ZX-Diagrams

• URL: http://arxiv.org/abs/2603.09580v1
• Date: Tue, 10 Mar 2026 12:26:09 GMT
• Title: ZX-Flow: A Flexible Criterion for Deterministic Computation with ZX-Diagrams
• Abstract summary: We introduce a new flow criterion called ZX-flow, formulated using a new type of decoration of a ZX-diagram we call Pauli semiwebs.<n>We show that ZX-flow is straightforwardly preserved by all Clifford rewrites and furthermore that a ZX-diagram has ZX-flow if and only if it is Clifford-equivalent to a graph-like ZX-diagram with Pauli flow.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Flow criteria are used to efficiently extract computations, either in the form of measurement patterns or quantum circuits, from ZX-diagrams. Existing criteria such as causal flow, generalised flow, and Pauli flow, were all originally formulated for graph states, so they require ZX-diagrams to be in a very particular graph-state-like form. This form is easily broken by applying basic ZX rules and makes establishing some desirable properties very complicated. Here, we introduce a new "ZX-native" flow criterion called ZX-flow, formulated using a new type of decoration of a ZX-diagram we call Pauli semiwebs. These are a generalisation of Pauli webs, which have recently been used extensively in reasoning about fault-tolerant computations in the ZX-calculus. We show that ZX-flow is straightforwardly preserved by all Clifford rewrites and furthermore that a ZX-diagram has ZX-flow if and only if it is Clifford-equivalent to a graph-like ZX-diagram with Pauli flow. Finally, we show that any diagram with ZX-flow can be readily interpreted either as a deterministic measurement-based computation or as a Clifford isometry followed by a sequence of Pauli exponentials. The latter can then be efficiently extracted to a quantum circuit.

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