发表机构
Centrum Wiskunde & Informatica (CWI); QuSoft; University of Technology Sydney（荷兰数学与计算机科学研究学会; QuSoft; 悉尼科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出高效算法，用于容错测试未知量子态是否接近乘积态，并利用随机着色与分块谱投影技术，改进了最近乘积态学习，所需副本数对 n 无关。
AI 中文摘要
我们解决了如下问题：测试一个未知的 $n$ 量子比特态 $\rho$ 是否在状态重叠度量下 $a$-接近某个乘积态，或者与任何乘积态都 $b$-远离。我们提供了一个时间高效的算法来解决这个问题，该算法需要未知态的 $n$ 无关的副本数。我们的随机着色论证表明，对于任何未知态，总是存在将 $[n]$ 划分为 $q$ 个部分的划分，使得相对于该划分的最近乘积态的重叠平方仅大一个加性因子 $O(1/q)$。这将问题简化为对 $q$ 个部分的容错测试，且局部维度可能增长。结合这一见解与分块谱投影论证，我们可以证明 Harrow & Montanaro 的乘积态测试的自然 $k$-副本推广提供了一个高效的容错测试器。我们使用相同的随机着色和分块谱投影技术来获得一个显著改进的最近乘积态学习算法。特别地，我们给出一个算法，该算法接受 $\widetilde{O}\big((nd)^2\big)\\, 2^{\widetilde{O}(1/\varepsilon^8)}$ 个未知态的副本，并产生一个 $\varepsilon$-近似最优的乘积态。该学习器的关键技术组件是 Bakshi 等人的高保真乘积态学习算法的量子比特变体，以及基于 Werner 的最优克隆信道的采样技术。
英文摘要
We address the problem of testing whether an unknown $n$-qudit state $ρ$ is $a$-close to a product state or $b$-far away from any product state, as measured in terms of state overlap. We provide a time-efficient algorithm to solve this problem that requires an $n$-independent number of copies of the unknown state. Our random coloring argument shows that for any unknown state there always exists a partition of $[n]$ into $q$ parts such that the square of the overlap with the closest product state with respect to this partition is only an additive factor $O(1/q)$ larger. This reduces the problem to tolerant testing of $q$ parties, with potentially growing local dimensions. Combining this insight with blockwise spectral projection arguments we can show that the natural $k$-copy generalization of Harrow & Montanaro's product state test provides an efficient tolerant tester. We use the same random coloring and blockwise spectral projection techniques to obtain a substantially improved algorithm for closest product state learning. In particular we give an algorithm that takes in $\widetilde{O}\big((nd)^2\big)\, 2^{\widetilde{O}(1/\varepsilon^8)}$ copies of the unknown state and produces an $\varepsilon$-approximately optimal product state. The key technical components of this learner are a qudit variant of the Bakshi et al. high-fidelity product state learning algorithm and a sampling technique based on Werner's optimal cloning channel.
