Local equivalence of stabilizer states: a graphical characterisation

2024-09-30

arXiv.org

AI Summary

The long-standing open problem of providing a complete graph-theoretic characterization of local unitary (LU) equivalence among graph states remains unresolved, hindered by the insufficiency of conventional local complementation. Method: We introduce generalized local complementation (GLC), a novel graph operation that fully captures LU equivalence for graph states. Leveraging the theory of minimal local sets, we integrate stabilizer formalism, graph transformations, and vertex typing to construct a canonical normal form. Contributions: We establish a strict infinite hierarchy of LU equivalence classes; derive a decidable graphical criterion for LU equivalence; prove invariance of minimal local sets under LU operations; and design a polynomial-time algorithm for LU-equivalence verification. These results provide foundational tools for quantum graph-state classification, measurement-basis reconstruction, and interdisciplinary research bridging graph theory and quantum information science.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Local Search
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

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.

Problem

• Quantum Graph States
• LU Equivalence
• Graph Representation

Innovation

• Generalized Local Completion
• Minimum Local Set
• Quantum Information Processing