Quantum entanglement is a crucial resource for learning properties from nature, but a precise characterization of its advantage can be challenging. In this Letter, we consider learning algorithms without entanglement to be those that only utilize states, measurements, and operations that are separable between the main system of interest and an ancillary system. Interestingly, we show that these algorithms are equivalent to those that apply quantum circuits on the main system interleaved with mid-circuit measurements and classical feedforward. Within this setting, we prove a tight lower bound for Pauli channel learning without entanglement that closes the gap between the best-known upper and lower bound. In particular, we show that $\mathrm{\Theta }(2^n{ϵ}^{-2})$ rounds of measurements are required to estimate each eigenvalue of an $n$-qubit Pauli channel to $ϵ$ error with high probability when learning without entanglement. In contrast, a learning algorithm with entanglement only needs $\mathrm{\Theta }({ϵ}^{-2})$ copies of the Pauli channel. The tight lower bound strengthens the foundation for an experimental demonstration of entanglement-enhanced advantages for Pauli noise characterization.

• Received 7 December 2023
• Revised 8 April 2024
• Accepted 15 April 2024

DOI:https://doi.org/10.1103/PhysRevLett.132.180805