Retrieving non-linear features from noisy quantum states

TL;DR

This paper develops efficient quantum protocols to extract high-order moments from noisy quantum states, demonstrating their practicality and the role of entanglement, with implications for quantum information tasks under realistic noise conditions.

Contribution

It introduces a novel, noise-invertible protocol with optimal sample complexity for high-order moment retrieval, utilizing observable shift and entanglement, scalable to large quantum systems.

Findings

Protocols work if and only if noise channel is invertible

Achieves lower overheads compared to conventional methods

Demonstrates entanglement's power in noisy high-order information retrieval

Abstract

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.