FET: Small: Foundations of Quantum State Learning and Testing

该项目的目标是推进我们对如何最有效地学习或测试量子粒子系统状态的科学理解。每当研究人员/工程师建造量子设备时——比如，用于量子隐形传态实验，或者作为量子计算机的一个组成部分——他们都需要检查它是否按预期工作。也就是说，他们需要能够确定它产生的粒子的量子态。如果从该装置中获得足够的样品，就有可能以任何所需的精度确定量子态。但由于量子实验的昂贵性质，寻求高效的策略来估计未知状态是很重要的，这些策略使用尽可能少的样本。该项目的总体目标是尝试在数学上确定最佳样本与精度权衡问题，如：a)学习未知量子态；B)测试未知状态是否等于期望的目标状态，等等。这个项目的成功将告诉我们一个效率基准，所有实际的状态学习方法都可以据此来判断。此外，该项目包含一个重要的教育组成部分，涉及本科生和研究生的研究工作，所获得的见解也将过滤到一个免费的在线量子计算课程中。在更技术性的层面上，该项目试图理解量子态断层扫描和估计中许多基本问题的最佳样本复杂性。提出的问题包括：a)使用o（d^2）个样本估计d维状态的特征值；B)量子层析成像和更严格的精度测量的状态认证，如卡方散度；C)使用o（d^4）个样本测试d × d量子态是可分离的（未纠缠的）还是远离可分离的；D)使用对数(m)为线性且维数为多对数的样本进行“m测量阴影断层扫描”；e)提高已知样本优化算法的计算效率。对于其中的许多问题，研究者计划使用并扩展最近与其他研究人员共同开发的基于表示理论的量子学习框架。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The goal of this project is to advance our scientific understanding of how to most efficiently learn or test the state of a quantum particle system. Whenever researchers/engineers build a quantum device -- say, for a quantum teleportation experiment, or as a component of a quantum computer -- they need to check whether it works as intended. That is, they need to be able to determine the quantum state of the particles it produces. Given enough samples from the device, it is possible to determine the quantum state to any desired accuracy. But due to the expensive nature of quantum experiments, it is important to seek highly efficient strategies for estimating the unknown state, ones that use as few samples as possible. The overarching goal of the project is to try to mathematically determine the optimal samples vs. accuracy tradeoff for problems such as: a) learning an unknown quantum state; b) testing whether an unknown state is equal to a desired target state, etc. Success in the project will tell us the efficiency benchmark against which all practical state-learning methodologies can be judged. Additionally, the project contains a significant educational component involving undergraduate and graduate students in the research effort, and the insights gained will also filter into a free online quantum computing course curriculum.At a more technical level, the project seeks to understand the optimal sample complexity of many basic problems in quantum state tomography and estimation. Proposed problems include: a) estimating a d-dimensional state's eigenvalues using o(d^2) samples; b) quantum tomography and state certification with respect to more stringent measures of accuracy, such as chi-squared divergence; c) testing whether a d x d quantum state is separable (unentangled) or far from separable using o(d^4) samples; d) performing 'm-measurement shadow tomography' with a number of samples linear in log(m) and polylogarithmic in the dimension; and e) improving the computational efficiency of known sample-optimal algorithms. For many of these problems, the investigator plans to use and extend the representation theory-based quantum learning framework developed recently with other researchers.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.