Direct Estimates and Confidence Intervals for Fidelity of Quantum States

量子力学是科学中关于物理世界最详细的理论，充分利用量子力学使量子计算机等新技术成为可能。在构建量子系统时，一个基本要求是验证实验者实际上已经构建了他们想要构建的东西。由于这个原因，实验家通过测量（固有的概率）来测试量子系统。这个项目的目的是以一种新的方式处理这些测量，以便对实际量子态与预期量子态的差异给出更准确和可靠的估计。这种更可靠的验证程序是构建容错量子计算机的重要一步，因此，该项目有助于创建有用的量子计算机。因此，该项目的更广泛影响包括量子计算的潜在影响，以及对科学进步的总体贡献。该项目的技术目标是高确定性地估计两个量子态之间的保真度；特别是，所提出的方法产生了一个严格的置信区间，其中保真度位于（在期望的失败概率内）。该方法在该领域是新颖的，因为它使用计算构建的统计量，并直接应用于原始数据，而不需要数据平均或全层析成像。该方法优于现有的直接方法和最大似然层析成像方法。所提出的工作的智力贡献将显示证明问题适合于一个非常具体的现有统计框架，推导估计器并解决相关的优化问题，并扩展理论和应用，例如，量子通道和纠缠见证。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。

Quantum mechanics is science's most detailed theory of the physical world, and fully exploiting quantum mechanics is enabling new technologies such as quantum computers. A fundamental requirement when building quantum systems is to verify that the experimentalist has actually built what they intended to build. For this reason, experimentalists test quantum systems by taking (inherently probabilistic) measurements. The aim of this project is to process these measurements in a new way in order to give a more accurate and reliable estimate of how much the actual quantum state differs from the intended quantum state. This more reliable verification procedure is an important step in constructing fault-tolerant quantum computers, and as such, this project contributes toward creating useful quantum computers. Thus the broader impact of the project includes the potential impacts of quantum computing, as well as generally contributing to the progress of science.The technical goal of the project is to estimate the fidelity between two quantum states with high certainty; in particular, the proposed method produces a rigorous confidence interval within which the fidelity lies (within a desired probability of failure). The approach is novel in the field, as it uses a computationally constructed statistic, and is applied directly to the raw data without requiring data-averaging or full tomography. The method has advantages over existing direct methods as well as over maximum likelihood tomography methods. The intellectual contributions of the proposed work will be showing that the certification problem fits inside a very specific existing statistical framework, deriving the estimators and solving related optimization problems, and extending both the theory and applications, e.g., to quantum channels and entanglement witnesses.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.