Motivated by estimation of quantum noise models, we study the problem of learning a Pauli channel, or more generally the Pauli error rates of an arbitrary channel. By employing a novel reduction to the "Population Recovery" problem, we give an extremely simple algorithm that learns the Pauli error rates of an n-qubit channel to precision ε in 𝓁_∞ using just O(1/ε²) log(n/ε) applications of the channel. This is optimal up to the logarithmic factors. Our algorithm uses only unentangled state preparation and measurements, and the post-measurement classical runtime is just an O(1/ε) factor larger than the measurement data size. It is also impervious to a limited model of measurement noise where heralded measurement failures occur independently with probability ≤ 1/4. We then consider the case where the noise channel is close to the identity, meaning that the no-error outcome occurs with probability 1-η. In the regime of small η we extend our algorithm to achieve multiplicative precision 1 ± ε (i.e., additive precision εη) using just O(1/(ε²η)) log(n/ε) applications of the channel.
@InProceedings{flammia_et_al:LIPIcs.TQC.2021.8, author = {Flammia, Steven T. and O'Donnell, Ryan}, title = {{Pauli Error Estimation via Population Recovery}}, booktitle = {16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021)}, pages = {8:1--8:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-198-6}, ISSN = {1868-8969}, year = {2021}, volume = {197}, editor = {Hsieh, Min-Hsiu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2021.8}, URN = {urn:nbn:de:0030-drops-140034}, doi = {10.4230/LIPIcs.TQC.2021.8}, annote = {Keywords: Pauli channels, population recovery, Goldreich-Levin, sparse recovery, quantum channel tomography} }
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