发表机构
University of Copenhagen（哥本哈根大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究噪声输入下弱Schur采样的鲁棒性，证明副本误差不随输入规模累积，并首次证明Keyl-Werner量子谱估计算法具有噪声鲁棒性。
AI 中文摘要
我们开创了对噪声输入下弱Schur采样（WSS）的研究。许多最优量子学习算法依赖于这种测量，例如在量子谱估计（QSE）和量子态层析（QST）等基础任务中。WSS、QSE和QST的标准分析假设目标态$\rho$的$n$个输入副本是相同的。我们研究当它们不相同时会发生什么，这是一种更现实的噪声场景，它打破了WSS所依赖的排列对称性。在最简单的噪声模型中，副本是独立的但不一定相同，即它们形成乘积态，并且每个副本在迹距离上与$d$维目标态$\rho$的$\epsilon$-接近。一个天真的数据处理论证让每个副本的误差在测量输出上累积为$n\epsilon$，随输入规模增长。我们证明它们不会累积：期望输出误差在总变差距离上至多为$\epsilon+\eta(n,d)$，其中$\eta(n,d)=O(d/\sqrt n)$是无噪声误差，加性噪声项$\epsilon$是最优的。在噪声乘积态之外，对于任意输入$\omega$，我们证明WSS的结果可观测量相对于De Palma等人的量子Wasserstein距离$W_1$是$1/n$-Lipschitz的，因此误差至多为$\eta(n,d)+\frac{1}{n}\\|\omega-\rho^{\otimes n}\\|_{W_1}$。特别地，我们的结果首次意味着Keyl和Werner关于QSE的开创性算法在提出二十多年后具有噪声鲁棒性。在我们的鲁棒性模型中，承诺不能弱化为单副本边缘分布的接近性或仅全局迹距离预算，因为在两种情况下某些噪声输入会击败所有测量和估计器。我们的结果是实践中最优量子学习的必要步骤，并有助于推动量子信息理论中许多其他问题的鲁棒性进展。
英文摘要
We initiate the study of weak Schur sampling (WSS) under noisy inputs. Many optimal quantum learning algorithms rely on this measurement, for instance in quantum spectrum estimation (QSE) and quantum state tomography (QST). The standard analysis of WSS, QSE and QST assumes that the $n$ input copies of the target state $ρ$ are identical, i.e. $ρ^{\otimes n}$. We study what happens when they are not, a more realistic noisy scenario that breaks the very permutation symmetry on which WSS is built. In the simplest model the input is a noisy product state: the copies are independent but not necessarily identical and each is $ε$-close in trace distance to the $d$-dimensional target state $ρ$. A naive argument via the data-processing inequality allows the per-copy errors to accumulate to $nε$ on the measurement outcome. We show that for QSE via WSS this does not happen: the expected estimation error in total variation is at most $ε+η(n,d)$, where $η(n,d)=O(d/\sqrt n)$ is the noiseless error, and the additive noise term $ε$ is optimal. The output error does not grow with $n$ even though the input error does; in this sense WSS is robust. Beyond noisy product states, for arbitrary input $ω$ we show that the error is at most $η(n,d)+\frac{1}{n}\|ω-ρ^{\otimes n}\|_{W_1}$, where $W_1$ is the quantum Wasserstein distance of De Palma et al. In particular, our results imply, for the first time, noise-robustness of Keyl and Werner's seminal algorithm for QSE, more than two decades after its introduction. Finally, we show that the noise cannot be too bad: within our model of robustness the promise on the input cannot be weakened to closeness of single-copy marginals or to a global trace-distance budget alone, as in both cases some inputs defeat every measurement and estimator.
Commentsv2: minor corrections in the phrasing of the results, abstract revised
