Tight Bounds on Pauli Channel Learning without Entanglement

Creators: Chen, Senrui; Oh, Changhun; Zhou, Sisi; Huang, Hsin-Yuan; Jiang, Liang

Abstract

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

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Acknowledgement

We thank Matthias Caro, Steve Flammia, John Preskill, and Alireza Seif for helpful discussions. We thank Sitan Chen and Weiyuan Gong for communicating with us about their independent and contemporaneous results. S. C., C. O., and L. J. acknowledge support from the ARO (W911NF-23-1-0077), ARO MURI (W911NF-21-1-0325), AFOSR MURI (FA9550-19-1-0399, FA9550-21-1-0209), NSF (OMA-1936118, ERC-1941583, OMA-2137642), NTT Research, and the Packard Foundation (2020-71479). This material is based upon work supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers. S. C. thanks Caltech IQIM for hospitality, where part of this work was completed. S. Z. acknowledges funding provided by the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center (NSF Grant PHY-1733907) and Perimeter Institute for Theoretical Physics, a research institute supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Colleges and Universities. H. H. is supported by a Google Ph.D. fellowship and a MediaTek Research Young Scholarship. H. H. acknowledges the visiting associate position at Massachusetts Institute of Technology.

Data Availability

The supplemental PDF file contains additional details about the proof and calculation that support the results stated in the letter.

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