Title: Expectation value estimation with parametrized quantum circuits Show Chinese title

Title: 参数化量子电路的期望值估计

Abstract: Estimating properties of quantum states, such as fidelities, molecular energies, and correlation functions, is a fundamental task in quantum information science. Due to the limitation of practical quantum devices, including limited circuit depth and connectivity, estimating even linear properties encounters high sample complexity. 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 two decomposition algorithms, a tensor network approach and a greedy projection approach that decompose 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 expectation values of some specific Hamiltonians and inner product of a Slater determinant with a pure state, highlighting 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。 由于实际量子设备的限制，包括有限的电路深度和连接性，即使估计线性性质也会遇到高样本复杂度的问题。 为了解决这种低效问题，我们提出了一种框架，该框架优化了使用浅层参数化量子电路估计任意可观测量期望值的样本复杂度。 在该框架内，我们引入了两种分解算法，一种是张量网络方法，另一种是贪心投影方法，这些方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。 我们通过估计一些特定哈密顿量的期望值以及单行列式与纯态的内积，数值上展示了我们算法的性能，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。