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

Aleks Kissinger, John van de Wetering

TL;DR

It is shown that ZX-flow is straightforwardly preserved by all Clifford rewrites and furthermore 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.

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.

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

TL;DR

It is shown that ZX-flow is straightforwardly preserved by all Clifford rewrites and furthermore 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.

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.
Paper Structure (16 sections, 16 theorems, 10 equations, 2 figures)

This paper contains 16 sections, 16 theorems, 10 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. ZX-diagrams
  4. Pauli webs
  5. Pauli Flow
  6. Pauli Semiwebs
  7. ZX-flow
  8. Clifford preservation of ZX-flow
  9. Equivalence of ZX-Flow and Pauli flow
  10. Circuit extraction
  11. Conclusion
  12. Correctness of the Definition of Pauli flow
  13. Semiweb Proofs
  14. Focusing ZX-flow
  15. Clifford Preservation Proofs
  16. ...and 1 more sections

Key Result

Theorem 2.4

Let $\vec{w}$ be a Pauli web for a ZX-diagram $D$ which colours the inputs of $D$ according to a Pauli string $\vec{P}$ and the outputs according to $\vec{Q}$. Then we have $D = (-1)^k \vec{Q} D \vec{P}$.

Figures (2)

  • Figure 1: Left: Example of a basic semiweb. Note that all the spiders on the boundary of the web have a $\pi$-defect. Middle: different possibilities for an edge semiweb. Right: decomposing a Pauli semiweb into a basic semiweb (top) and edge semiwebs (bottom).
  • Figure 2: The extended Clifford ZX-calculus, where $\alpha \in \mathbb R$, $k \in \mathbb Z$, and $c = \sqrt{2}^{(n-1)(m-1)}$ for $n$ inputs and $m$ outputs. The rules are pronounced: (sp) spider, (cc) colour-change, ($\pi$) $\pi$ stabiliser, (sc) strong complementarity, (h) Hadamard, (id) identity, (s1) and (s2) scalar rules.

Theorems & Definitions (43)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • proof
  • Example 2.5: Firing a Pauli web
  • Theorem 2.6
  • Theorem 2.7
  • proof
  • Example 2.8: Pauli webs for a unitary
  • ...and 33 more