Tight bounds on Pauli channel learning without entanglement

TL;DR

This paper establishes a tight lower bound on the number of measurements needed to learn Pauli channels without entanglement, highlighting the significant advantage of entanglement in quantum channel estimation.

Contribution

It provides the first tight lower bound for Pauli channel learning without entanglement, clarifying the fundamental limits and advantages of entanglement in quantum noise characterization.

Findings

Learning without entanglement requires exponentially more measurements than with entanglement.

The lower bound matches the best known upper bound, confirming its tightness.

Entanglement provides a polynomial speedup in Pauli channel learning.

Abstract

Quantum entanglement is a crucial resource for learning properties from nature, but a precise characterization of its advantage can be challenging. In this work, 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 $\Theta(2^n\varepsilon^{-2})$ rounds of measurements are required to estimate each eigenvalue of an $n$-qubit Pauli channel to $\varepsilon$ error with high probability when learning without entanglement. In contrast, a learning algorithm with entanglement only needs $\Theta(\varepsilon^{-2})$ copies of the Pauli channel. The tight lower bound strengthens the foundation for an experimental demonstration of entanglement-enhanced advantages for Pauli noise characterization.