Polynomial-time local-unitary equivalence of graph states

2026-09-30

AI Summary

This study addresses the long-standing open problem of determining local unitary (LU) equivalence of graph states in quantum information theory. We propose the first polynomial-time deterministic algorithm for this task, which replaces conventional vertex subset enumeration with a compact constraint system. By integrating linear equation solving over binary fields, graph preprocessing, and constraint generation techniques, the proposed method efficiently determines LU equivalence and constructs the corresponding single-qubit unitary operations. This work reduces the computational complexity to O(n^6.38). Furthermore, the framework is extensible to computing the number of local Clifford (LC) equivalence classes and deciding the equivalence of stabilizer codes.

Abstract

Local-unitary (LU) equivalence asks whether two quantum states differ only by independent changes of basis on their qubits. For graph states, whether this relation can be decided in polynomial time has remained open for over a decade. We give a deterministic algorithm that decides LU equivalence for graphs on $n$ labelled vertices in $\widetilde O(n^{6.38})$ bit operations and constructs exact single-qubit unitaries whenever the states are equivalent. Building on Claudet and Perdrix's quasipolynomial algorithm, we replace the enumeration of vertex subsets by a compact system of constraints generated from pairs and triples. The remaining graph transformation is found by solving linear equations over the binary field. These new steps cost $\widetilde O(n^5)$ bit operations; the inherited graph preprocessing sets the overall bound. We also count the local-Clifford (LC) classes of graph states within any LU class: their number is a power of two, computable within the same bound. For any given graph state, this decides whether single-qubit Clifford gates reach every graph state in its LU class, and supplies a counterexample when they do not. The method also decides LU equivalence of stabilizer codes encoding one logical qubit.

Problem

• Local-unitary equivalence
• Graph states
• Polynomial time
• Stabilizer codes
• Local-Clifford classes

Innovation

• Local-unitary equivalence
• Graph states
• Polynomial-time algorithm
• Stabilizer codes
• Local-Clifford classes