  Accurately estimating high-order moments of quantum states is an elementary
precondition for many crucial tasks in quantum computing, such as entanglement
spectroscopy, entropy estimation, spectrum estimation, and predicting
non-linear features from quantum states. But in reality, inevitable quantum
noise prevents us from accessing the desired value. In this paper, we address
this issue by systematically analyzing the feasibility and efficiency of
extracting high-order moments from noisy states. We first show that there
exists a quantum protocol capable of accomplishing this task if and only if the
underlying noise channel is invertible. We then establish a method for deriving
protocols that attain optimal sample complexity using quantum operations and
classical post-processing only. Our protocols, in contrast to conventional
ones, incur lower overheads and avoid sampling different quantum operations due
to a novel technique called observable shift, making the protocols strong
candidates for practical usage on current quantum devices. The proposed method
also indicates the power of entangled protocols in retrieving high-order
information, whereas in the existing methods, entanglement does not help. We
further construct the protocol for large quantum systems to retrieve the
depolarizing channels, making the proposed method scalable. Our work
contributes to a deeper understanding of how quantum noise could affect
high-order information extraction and provides guidance on how to tackle it.
