发表机构
University of California, Los Angeles; Mani L. Bhaumik Institute for Theoretical Physics（加州大学洛杉矶分校; 马尼·L·巴乌米克理论物理研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出一种基于哈密顿量演化的高效量子算法制备对称群特征标态，门复杂度优于量子傅里叶变换方法，并推广到量子特征标变换，应用于共形场论纠缠熵。
AI 中文摘要
考虑一个量子态矢量，它与对称群（$S_n$）特征标表的任意给定列成正比，对应于由特征标值加权的不可约表示上的叠加。最近有人论证，在合理的复杂性理论假设下，从该态采样在经典上是困难的，而存在一种利用 $S_n$ 上的量子傅里叶变换（QFT）来制备该特征标态的高效量子算法 [arXiv:2501.12579]。我们给出了一种替代的、更简单的算法，通过一系列哈密顿量演化来制备该特征标态，并提供了一种使用可重构量子比特的高效实现，该实现仅使用一维中的最近邻门。计入哈密顿量模拟误差的门复杂度取决于列的选择：对于固定的目标态误差，估计的充分总门数范围从 $n$-循环列的最少 $O(n)$ 个门，到推测为经典困难的参数区域附近的 $\widetilde O(n^{1.75})$ 个门，再到最坏情况下可证明的 $\widetilde O(n^{2.5})$ 个门。相比之下，先前方法中的 QFT 子程序使用 $\widetilde O(n^3)$ 个门。我们的门数估计来自领先阶的态依赖 Trotter 误差分析，而可证明的最坏情况界依赖于基于 QSVT 的哈密顿量模拟方法；基于 Trotter 协议的数值基准表明这些估计是保守的。此外，我们将我们的协议推广，给出了一个使用 $\widetilde O(n^{2.5})$ 个门和 $O(n)$ 个量子比特的 $S_n$ 上量子特征标变换（QCT）的显式算法。还讨论了其在对称轨道共形场论的纠缠熵中的应用。
英文摘要
Consider a quantum state vector proportional to any given column of the character table of the symmetric group ($S_n$), corresponding to a superposition over the irreps weighted by the character values. It was recently argued that sampling from this state is classically hard under reasonable complexity-theoretic assumptions, while there exists an efficient quantum algorithm for preparing this character state using the quantum Fourier transform (QFT) over $S_n$ [arXiv:2501.12579]. We give an alternative, simpler algorithm to prepare this character state with a sequence of Hamiltonian evolutions, as well as an efficient implementation with reconfigurable qubits that uses only nearest-neighbor gates in 1D. The gate complexity, accounting for Hamiltonian simulation errors, depends on the choice of the column: For fixed target state error, the estimated sufficient total gate count ranges from as few as $O(n)$ gates for the $n$-cycle column, to $\widetilde O(n^{1.75})$ gates near the conjecturally classically hard parameter regime, to provably $\widetilde O(n^{2.5})$ gates in the worst case. In contrast, the QFT subroutine in the prior approach uses $\widetilde O(n^3)$ gates. Our gate count estimates come from leading-order state-dependent Trotter error analysis, and the proven worst-case bound relies on a QSVT-based Hamiltonian simulation method; numerical benchmarks of the Trotter-based protocol suggest that the estimates are conservative. Furthermore, we generalize our protocol to give an explicit algorithm for the quantum character transform (QCT) over $S_n$ using $\widetilde O(n^{2.5})$ gates and $O(n)$ qubits. An application to entanglement entropy of symmetric orbifold conformal field theories is also discussed.
Comments54 pages, 8 figures
