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Local Equivalence of Stabilizer States: A Graphical Characterisation

Authors Nathan Claudet , Simon Perdrix

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Author Details

Nathan Claudet
  • Inria Mocqua, LORIA, CNRS, Université de Lorraine, F-54000 Nancy, France
Simon Perdrix
  • Inria Mocqua, LORIA, CNRS, Université de Lorraine, F-54000 Nancy, France

Funding

This work is supported by the the Plan France 2030 through the PEPR integrated project EPiQ ANR-22- PETQ-0007 and the HQI platform ANR-22-PNCQ-0002; and by the European projects NEASQC and HPCQS.

Acknowledgements

The authors want to thank David Cattaneo and Mehdi Mhalla for fruitful discussions on previous versions of this paper.

Cite As Get BibTex

Nathan Claudet and Simon Perdrix. Local Equivalence of Stabilizer States: A Graphical Characterisation. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 27:1-27:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.STACS.2025.27

Abstract

Stabilizer states form a ubiquitous family of quantum states that can be graphically represented through the graph state formalism. A fundamental property of graph states is that applying a local complementation - a well-known and extensively studied graph transformation - results in a graph that represents the same entanglement as the original. In other words, the corresponding graph states are LU-equivalent. This property served as the cornerstone for capturing non-trivial quantum properties in a simple graphical manner, in the study of quantum entanglement but also for developing protocols and models based on graph states and stabilizer states, such as measurement-based quantum computing, secret sharing, error correction, entanglement distribution... However, local complementation fails short to fully characterise entanglement: there exist pairs of graph states that are LU-equivalent but cannot be transformed one into the other using local complementations. Only few is known about the equivalence of graph states beyond local complementation. We introduce a generalisation of local complementation which graphically characterises the LU-equivalence of graph states. We use this characterisation to show the existence of a strict infinite hierarchy of equivalences of graph states. Our approach is based on minimal local sets, which are subsets of vertices that are known to cover any graph, and to be invariant under local complementation and even LU-equivalence. We use these structures to provide a type to each vertex of a graph, leading to a natural standard form in which the LU-equivalence can be exhibited and captured by means of generalised local complementation.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum information theory
Keywords
  • Quantum computing
  • Graph theory
  • Entanglement
  • Local complementation

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