Optimization for expectation value estimation with shallow quantum circuits

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.