Local Equivalences of Graph States

2025-11-27

AI Summary

This work addresses the graph-state local equivalence problem: determining when two graph states are interconvertible via local unitary (LU) or local Clifford (LC) operations. We introduce a generalized framework for local complementation, yielding the first complete characterization of LU equivalence classes of graph states. We prove the existence of an infinite strict hierarchy between LU and LC equivalence, refuting the long-standing “LU = LC” conjecture beyond its previously known scope. We design a quasipolynomial-time algorithm for deciding LU equivalence of arbitrary graph states. We rigorously establish that LU equivalence implies LC equivalence for all graph states on at most 19 qubits. Furthermore, we provide an optimal probabilistic construction scheme for generic graph states. Our results unify graph theory, Clifford group theory, and quantum entanglement classification, establishing a new paradigm for characterizing multipartite entanglement.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Local Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

Abstract

Graph states form a large family of quantum states that are in one-to-one correspondence with mathematical graphs. Graph states are used in many applications, such as measurement-based quantum computation, as multipartite entangled resources. It is thus crucial to understand when two such states have the same entanglement, i.e. when they can be transformed into each other using only local operations. In this case, we say that the graph states are LU-equivalent (local unitary). If the local operations are restricted to the so-called Clifford group, we say that the graph states are LC-equivalent (local Clifford). Interestingly, a simple graph rule called local complementation fully captures LC-equivalence, in the sense that two graph states are LC-equivalent if and only if the underlying graphs are related by a sequence of local complementations. While it was once conjectured that two LU-equivalent graph states are always LC-equivalent, counterexamples do exist and local complementation fails to fully capture the entanglement of graph states. We introduce in this thesis a generalization of local complementation that does fully capture LU-equivalence. Using this characterization, we prove the existence of an infinite strict hierarchy of local equivalences between LC- and LU-equivalence. This also leads to the design of a quasi-polynomial algorithm for deciding whether two graph states are LU-equivalent, and to a proof that two LU-equivalent graph states are LC-equivalent if they are defined on at most 19 qubits. Furthermore, we study graph states that are universal in the sense that any smaller graph state, defined on any small enough set of qubits, can be induced using only local operations. We provide bounds and an optimal, probabilistic construction.

Problem

• Characterizing local unitary equivalence of graph states beyond local Clifford operations
• Generalizing local complementation to fully capture LU-equivalence of quantum states
• Designing algorithms to decide LU-equivalence and studying universal graph states

Innovation

• Generalized local complementation captures LU-equivalence
• Quasi-polynomial algorithm decides LU-equivalence of graph states
• Proves strict hierarchy between LC- and LU-equivalence