以高精度和低样本复杂度预测Scrooge系综的性质
Predicting properties of Scrooge ensembles with high accuracy and low sample complexity
浏览论文内容
中文总结 AI 辅助
本文提出多项式近似与高效量子算法，以指数级误差改进和低样本复杂度预测Scrooge系综的高阶矩性质，并实现对称子空间投影，避免昂贵后选择。
中文摘要 AI 辅助
与密度矩阵$\rho$相关联的Scrooge系综是平均密度矩阵为$\rho$的最大随机系综。因其在量子多体动力学和量子信息中的应用而受到越来越多的关注。Scrooge系综的内在统计性质编码在其高阶矩中，然而这些矩由于具有非多项式Haar被积函数而难以分析和制备，这阻碍了Haar积分机制的直接使用。同时，现有的从Scrooge系综中采样多个状态副本的物理机制通常需要昂贵的后选择，导致其在大系统尺寸下效率低下。在本工作中，我们开发了分析和实现Scrooge矩的高效且准确的方法。我们首先引入对Scrooge $k$阶矩的多项式近似，其误差可调至一个阈值，该阈值相比先前结果在$1/||\rho||_\infty$上从逆多项式改进为指数级。基于我们的近似，我们随后给出了第一个高效量子算法，该算法使用$\widetilde{\mathcal{O}}(k^3/\epsilon^5)$个$\rho$的副本即可估计可观测量期望值，适用于该阈值以上的任意目标误差$\epsilon$。我们还进一步提供了当$\rho$的制备电路已知时Scrooge $k$阶矩的块编码构造。我们算法中的一个关键子程序可能具有独立意义，它使用$\mathcal{O}(k^3/\epsilon^2)$个$\rho$样本高效实现$\rho^{\otimes k}$到$k$重对称子空间的投影，避免了该投影的朴素$k!$后选择成本。这些结果为Scrooge系综性质的理论分析和实际预测提供了新工具，并有望应用于量子多体动力学和量子信息处理。
英文摘要
The Scrooge ensemble associated with a density matrix $ρ$ is the maximally random ensemble whose average density matrix is $ρ$. It has attracted growing interest for its applications to quantum many-body dynamics and quantum information. The intrinsic statistical properties of the Scrooge ensembles are encoded in their higher-order moments, yet they are nontrivial to analyze and prepare for they feature a non-polynomial Haar integrand, blocking direct use of Haar integration machinery. Existing physical mechanisms to sample multiple copies of states from the Scrooge ensembles, meanwhile, typically require costly postselection, rendering them inefficient for large system size. In this work, we develop efficient and accurate methods for analyzing and implementing Scrooge moments. We first introduce a polynomial approximation to the Scrooge $k$-th moments whose error is tunable down to a threshold that improves from inverse polynomial to exponential in $1/||ρ||_\infty$ over previous results. Building on our approximation, we then give the first efficient quantum algorithm that estimates observable expectation values using $\widetilde{\mathcal{O}}(k^3/ε^5)$ copies of $ρ$ for any target error $ε$ above this threshold. We further provide a construction of a block encoding of the Scrooge $k$-th moment when the preparation circuit of $ρ$ is known. Of independent interest could be a key subroutine in our algorithms, which efficiently implements projection onto the $k$-fold symmetric subspace for $ρ^{\otimes k}$ using $\mathcal{O}(k^3/ε^2)$ samples of $ρ$, avoiding the naive $k!$ postselection cost of such a projection. These results provide new tools for both the theoretical analysis and practical prediction of properties of Scrooge ensembles, with potential applications to quantum many-body dynamics and quantum information processing.
发表机构
- Chen-Ning Yang Institute for Advanced Study, Tsinghua University（清华大学杨振宁高等研究院）
- Beijing Key Laboratory of Cold Atom Quantum Computation, Tsinghua University（清华大学冷原子量子计算北京市重点实验室）
- Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院物理研究所）
- School of Computer and Communication Sciences, École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院计算机与通信科学学院）
机构由 AI 辅助整理，请以论文原文为准。
