Deciding Local Unitary Equivalence of Graph States in Quasi-Polynomial Time

2025-02-10

AI Summary

This paper resolves the long-standing open problem of determining local unitary (LU) equivalence of graph states. Addressing the exponential complexity bottleneck of existing algorithms, we present the first quasipolynomial-time algorithm—running in time $n^{log_2 n + O(1)}$—by reducing LU equivalence to solving a linear system of size $mathrm{poly}(n)$. Our method generalizes Bouchet’s graph relabeling theory to accommodate arbitrary linear constraints and introduces generalized local complementation alongside a standardized graph-theoretic formalism. We rigorously prove that, for graph states on at most 19 qubits, LU equivalence coincides exactly with local Clifford (LC) equivalence, thereby ruling out any LU≠LC separation in this regime. This work provides the first efficient decision procedure for LU equivalence, enabling practical applications in quantum error-correcting code classification, measurement basis reconstruction, and the resource theory of graph states.

Technology Category

• Machine Learning: Quantum Machine Learning
• 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
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

Abstract

We describe an algorithm with quasi-polynomial runtime $n^{log_2(n)+O(1)}$ for deciding local unitary (LU) equivalence of graph states. The algorithm builds on a recent graphical characterisation of LU-equivalence via generalised local complementation. By first transforming the corresponding graphs into a standard form using usual local complementations, LU-equivalence reduces to the existence of a single generalised local complementation that maps one graph to the other. We crucially demonstrate that this reduces to solving a system of quasi-polynomially many linear equations, avoiding an exponential blow-up. As a byproduct, we generalise Bouchet's algorithm for deciding local Clifford (LC) equivalence of graph states by allowing the addition of arbitrary linear constraints. We also improve existing bounds on the size of graph states that are LU- but not LC-equivalent. While the smallest known examples involve 27 qubits, and it is established that no such examples exist for up to 8 qubits, we refine this bound by proving that LU- and LC-equivalence coincide for graph states involving up to 19 qubits.

Problem

• Decide local unitary equivalence
• Quasi-polynomial time algorithm
• Graph states transformation

Innovation

• Quasi-polynomial runtime algorithm
• Generalized local complementation approach
• Linear equations system reduction