The 27-qubit Counterexample to the LU-LC Conjecture is Minimal

2026-03-26

AI Summary

This study investigates the minimal counterexample to the LU-LC conjecture, which posits whether there exists a graph state on fewer than 27 qubits that is locally unitary (LU) equivalent but not locally Clifford (LC) equivalent. By analyzing the structure of -local complementation operations and leveraging algebraic connections between triorthogonal codes and Reed–Muller codes, the authors systematically examine all graph states on up to 26 qubits. Their analysis demonstrates that, within this range, LU equivalence coincides exactly with LC equivalence. Consequently, this work establishes for the first time that any counterexample to the LU-LC conjecture must involve at least 27 qubits, thereby precisely delineating the boundary where LU-LC equivalence holds.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Local Search
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

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

Abstract

It was once conjectured that two graph states are local unitary (LU) equivalent if and only if they are local Clifford (LC) equivalent. This so-called LU-LC conjecture was disproved in 2007, as a pair of 27-qubit graph states that are LU-equivalent, but not LC-equivalent, was discovered. We prove that this counterexample to the LU-LC conjecture is minimal. In other words, for graph states on up to 26 qubits, the notions of LU-equivalence and LC-equivalence coincide. This result is obtained by studying the structure of 2-local complementation, a special case of the recently introduced r-local complementation, and a generalization of the well-known local complementation. We make use of a connection with triorthogonal codes and Reed-Muller codes.

Problem

• LU-LC conjecture
• graph states
• local unitary equivalence
• local Clifford equivalence
• minimal counterexample

Innovation

• LU-LC conjecture
• graph states
• local complementation
• triorthogonal codes
• Reed-Muller codes