Title: Optimization for expectation value estimation with shallow quantum circuits Show Chinese title

Title: 浅层量子电路的期望值估计优化

Abstract: Estimating linear properties of quantum states, such as fidelities, molecular energies, and correlation functions, is a fundamental task in quantum information science. The classical shadow has emerged as a prevalent tool due to its efficiency in estimating many independent observables simultaneously. However, it does not utilize the information of the target observable and the constraints of quantum devices, making it inefficient in many practical scenarios where the focus is on estimating a select few observables. To address this inefficiency, we propose a framework that optimizes sample complexity for estimating the expectation value of any observable using a shallow parameterized quantum circuit. Within this framework, we introduce a greedy algorithm that decomposes the target observable into a linear combination of multiple observables, each of which can be diagonalized with the shallow circuit. Using this decomposition, we then apply an importance sampling algorithm to estimate the expectation value of the target observable. We numerically demonstrate the performance of our algorithm by estimating the ground energy of a sparse Hamiltonian and the inner product of two pure states, highlighting the advantages compared to some conventional methods. Additionally, we derive the fundamental lower bound for the sample complexity required to estimate a target observable using a given shallow quantum circuit, thereby enhancing our understanding of the capabilities of shallow circuits in quantum learning tasks. Show Chinese abstract

Abstract: 估计量子态的线性性质，如保真度、分子能量和关联函数，是量子信息科学中的基本任务。 经典影子作为一种普遍工具脱颖而出，因为它在同时估计多个独立可观测量方面具有效率。 然而，它没有利用目标可观测量的信息和量子设备的约束，在许多实际场景中，重点是估计少数几个可观测量，这使得它效率低下。 为了解决这种低效问题，我们提出了一个框架，该框架优化了使用浅层参数化量子电路估计任何可观测量期望值的样本复杂度。 在此框架内，我们引入了一种贪心算法，将目标可观测量分解为多个可观测量的线性组合，每个可观测量都可以通过浅层电路对角化。 利用这种分解，我们随后应用重要性采样算法来估计目标可观测量的期望值。 我们通过估计稀疏哈密顿量的基态能量和两个纯态的内积来数值演示了我们的算法性能，突显了与一些传统方法相比的优势。 此外，我们推导了使用给定的浅层量子电路估计目标可观测量所需的样本复杂度的基本下限，从而增强了我们对浅层电路在量子学习任务中能力的理解。