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ZX-Flow: A Flexible Criterion for Deterministic Computation with ZX-Diagrams
Quantum Physics · Submitted 10 Mar 2026 · Published version
abstract compiled · 2026-11-05 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Flow criteria are used to efficiently extract computations, either in the form of measurement patterns or quantum circuits, from ZX-diagrams. | background 1.00 | 0.05 | 0.04 | asserted 1.00 |
| s1 | 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. | background 0.85 | 0.04 | 0.03 | asserted 1.00 |
| s2 | This form is easily broken by applying basic ZX rules and makes establishing some desirable properties very complicated. | background 0.71 | 0.09 | 0.02 | asserted 1.00 |
| s3 | 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. | contribution 1.00 | 0.94 | 0.21 | asserted 1.00 |
| s4 | These are a generalisation of Pauli webs, which have recently been used extensively in reasoning about fault-tolerant computations in the ZX-calculus. | background 0.89 | 0.10 | 0.03 | asserted 1.00 |
| s5 | 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. | result 0.98 | 0.95 | 0.41 | asserted 1.00 |
| s6 | 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. | result 0.80 | 0.95 | 0.33 | asserted 1.00 |
| s7 | The latter can then be efficiently extracted to a quantum circuit. | result 0.46 | 0.77 | 0.11 | asserted 0.95 |
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Original 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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ZX-calculus has become a widely used tool in quantum information and quantum computing, with applications ranging from quantum circuit optimization [ref] , quantum error correction [ref] , fault-tolerance [ref] , measurement-based quantum computation [ref] , quantum chemistry [ref] , quantum machine learning [ref] , and condensed matter physics [ref] .
A ribbon ZX calculus for gauge theory · Introduction · 0.87
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