Tight bounds on Pauli channel learning without entanglement

TL;DR

This Letter proves a tight lower bound for Pauli channel learning without entanglement that closes the gap between the best-known upper and lower bound and strengthens the foundation for an experimental demonstration of entanglement-enhanced advantages for Pauli noise characterization.

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 $Θ(2^n\varepsilon^{-2})$ 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 $Θ(\varepsilon^{-2})$ copies of the Pauli channel. The tight lower bound strengthens the foundation for an experimental demonstration of entanglement-enhanced advantages for Pauli noise characterization.

Figures (2)

• Figure 1: (a) Separable schemes. The two different colors indicate the operations are separable. (b) Classical-memory-assisted schemes. The double line represents classical registers. The square box represents adaptively-chosen (arrows coming from the classical registers) quantum instruments (outcomes sent to the classical registers). We will show the two schemes are equivalent in terms of sample complexity, so we call both entanglement-free schemes.
• Figure 2: Sample complexity for Pauli channel learning. The task is to estimate any Pauli fidelity to $\varepsilon=0.1$ additive precision with at least $2/3$ success probability. The dash lines represent our entanglement-free (EF) lower bound of Thm. \ref{['th:main']} and the ancilla-free lower bound from chen2022quantum. The solid lines represent the sample complexity upper bound calculated from an entanglement-assisted (EA) scheme with noisy Bell state and measurements. For simplicity, we assume the state preparation suffers from depolarizing noises, so that each noisy $2$-qubit Bell pair $\tilde{\rho}_\mathrm{Bell}$ has Fidelity $F_\mathrm{Bell}=\bra{\Psi_+}\tilde{\rho}_\mathrm{Bell}\ket{\Psi_+}$. The colored region indicates entanglement(ancilla)-enabled advantages.

Theorems & Definitions (14)

• Proposition 1
• Theorem 2
• proof : Proof Sketch of Theorem \ref{['th:main']}.
• Theorem 3
• Definition 1
• Definition 2
• Proposition 1
• proof
• Definition 3
• Theorem 2