发表机构
École Polytechnique Fédérale de Lausanne (EPFL); Princeton University（洛桑联邦理工学院; 普林斯顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一个确定性多项式时间算法，在$\widetilde O(n^{6.38})$位操作内判定图态的局部酉等价，并构造等价变换，同时计算局部Clifford类数量，适用于稳定子码。
AI 中文摘要
局部酉（LU）等价询问两个量子态是否仅在其量子比特上的独立基变换下不同。对于图态，这一关系是否能在多项式时间内判定已开放了十多年。我们给出一个确定性算法，在$\widetilde O(n^{6.38})$位操作内判定$n$个标记顶点上图态的LU等价，并在状态等价时构造精确的单量子比特酉算子。基于Claudet和Perdrix的拟多项式算法，我们将顶点子集的枚举替换为由二元组和三元组生成的紧凑约束系统。剩余的图变换通过求解二元域上的线性方程找到。这些新步骤花费$\widetilde O(n^5)$位操作；继承的图预处理设定了整体界。我们还计算任何LU类内图态的局部Clifford（LC）类：其数量是2的幂，可在相同界内计算。对于任何给定图态，这判定单量子比特Clifford门是否达到其LU类中的每个图态，并在未达到时提供反例。该方法还判定编码一个逻辑量子比特的稳定子码的LU等价。
英文摘要
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.
