  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.
