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

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 "ZXnative" 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 ZXflow 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 measurementbased computation or as a Clifford isometry followed by a sequence of Pauli exponentials.The latter can then be efficiently extracted to a quantum circuit.