Criteria for unbiased estimation: applications to noise-agnostic sensing and learnability of quantum channel

Abstract

We establish the necessary and sufficient conditions for unbiased estimation in multi-parameter estimation tasks. More specifically, we first consider quantum state estimation, where multiple parameters are encoded in a quantum state, and derive simple and intuitive necessary and sufficient conditions for an unbiased estimation based on the derivatives of the encoded state. To demonstrate the utility of our framework, we consider phase estimation under unknown Pauli noise. We show that while unbiased phase estimation is infeasible with a naive scheme, employing an entangled probe with a noiseless ancilla enables unbiased estimation. Next, we extend our analysis to quantum channel estimation (equivalently, quantum channel learning), where the goal is to estimate parameters characterizing an unknown quantum channel. We establish the necessary and sufficient condition for unbiased estimation of these parameters. Notably, by interpreting unbiased estimation as learnability, our result extends to the fundamental learnability of parameters in general quantum channels. As a concrete application, we investigate the learnability of noise affecting non-Clifford gates via cycle benchmarking.