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

TL;DR

ZX-Flow is a new criterion for deterministic quantum computation from ZX-diagrams, using Pauli semiweb decorations, that simplifies analysis and extraction of quantum circuits while being preserved under Clifford rewrites.

Contribution

Introduces ZX-flow, a ZX-native flow criterion based on Pauli semiwebs, generalizing previous flow concepts and compatible with Clifford transformations.

Findings

ZX-flow is preserved by Clifford rewrites.

A diagram with ZX-flow is Clifford-equivalent to a graph-like ZX-diagram with Pauli flow.

Diagrams with ZX-flow can be interpreted as measurement-based computation or as circuits with Pauli exponentials.

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.