Document Open Access Logo

Pauli Error Estimation via Population Recovery

Authors Steven T. Flammia, Ryan O'Donnell

Thumbnail PDF


  • Filesize: 0.67 MB
  • 16 pages

Document Identifiers

Author Details

Steven T. Flammia
  • AWS Center for Quantum Computing, Pasadena, CA, USA
  • IQIM, California Institute of Technology, Pasadena, CA, USA
Ryan O'Donnell
  • Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA


We thank Robin Harper for discussions about Pauli channels.

Cite AsGet BibTex

Steven T. Flammia and Ryan O'Donnell. Pauli Error Estimation via Population Recovery. In 16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 197, pp. 8:1-8:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


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.

Subject Classification

ACM Subject Classification
  • General and reference → Cross-computing tools and techniques
  • Hardware → Quantum error correction and fault tolerance
  • Pauli channels
  • population recovery
  • Goldreich-Levin
  • sparse recovery
  • quantum channel tomography


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Frank Ban, Xi Chen, Adam Freilich, Rocco Servedio, and Sandip Sinha. Beyond trace reconstruction: Population recovery from the deletion channel. In Proceedings of the 60th Annual IEEE Symposium on Foundations of Computer Science, pages 745-768, 2019. Google Scholar
  2. Frank Ban, Xi Chen, Rocco A. Servedio, and Sandip Sinha. Efficient Average-Case Population Recovery in the Presence of Insertions and Deletions. In Dimitris Achlioptas and László A. Végh, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019), volume 145 of Leibniz International Proceedings in Informatics (LIPIcs), pages 44:1-44:18, Dagstuhl, Germany, 2019. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. Google Scholar
  3. Lucia Batman, Russell Impagliazzo, Cody Murray, and Ramamohan Paturi. Finding heavy hitters from lossy or noisy data. In Proceedings of the 16th Annual International Conference on Approximation Algorithms for Combinatorial Optimization Problems, pages 347-362, 2013. Google Scholar
  4. Christoph Dankert. Efficient Simulation of Random Quantum States and Operators. PhD thesis, University of Waterloo, 2015. Google Scholar
  5. Anindya De, Ryan O'Donnell, and Rocco Servedio. Sharp bounds for population recovery. Theory of Computing, 16(6):1-20, 2020. Google Scholar
  6. Anindya De, Michael Saks, and Sijian Tang. Noisy population recovery in polynomial time. In Proceedings of the 57th Annual IEEE Symposium on Foundations of Computer Science, pages 675-684, 2016. Google Scholar
  7. Zeev Dvir, Anup Rao, Avi Wigderson, and Amir Yehudayoff. Restriction access. In Proceedings of the 3nd Annual Innovations in Theoretical Computer Science, pages 19-33, 2012. Google Scholar
  8. Steven Flammia and Joel Wallman. Efficient estimation of Pauli channels. ACM Transactions on Quantum Computing, 1(1):1-32, 2020. Google Scholar
  9. Oded Goldreich and Leonid Levin. A hard-core predicate for all one-way functions. In Proceedings of the 21st Annual ACM Symposium on Theory of Computing, pages 25-32, 1989. Google Scholar
  10. Robin Harper, Steven T. Flammia, and Joel J. Wallman. Efficient learning of quantum noise. Nature Physics, 16(12):1184–1188, August 2020. Google Scholar
  11. Robin Harper, Wenjun Yu, and Steven T. Flammia. Fast estimation of sparse quantum noise. PRX Quantum, 2(1):010322, February 2021. Google Scholar
  12. Jonas Helsen, Xiao Xue, Lieven M. K. Vandersypen, and Stephanie Wehner. A new class of efficient randomized benchmarking protocols. npj Quantum Information, 5(1):71, 2019. Google Scholar
  13. E. Knill. Quantum computing with realistically noisy devices. Nature, 434(7029):39-44, March 2005. Google Scholar
  14. Shachar Lovett and Jiapeng Zhang. Improved noisy population recovery, and reverse Bonami-Beckner inequality for sparse functions. In Proceedings of the 47th Annual ACM Symposium on Theory of Computing, pages 137-142, 2015. Google Scholar
  15. Shachar Lovett and Jiapeng Zhang. Noisy population recovery from unknown noise. In Conference on Learning Theory, pages 1417-1431, 2017. Google Scholar
  16. Ankur Moitra and Michael Saks. A polynomial time algorithm for lossy population recovery. In Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science, pages 110-116, 2013. Google Scholar
  17. Shyam Narayanan. Improved algorithms for population recovery from the deletion channel. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1259-1278. Society for Industrial and Applied Mathematics, 2021. Google Scholar
  18. Ryan O'Donnell. Analysis of Boolean functions. Cambridge University Press, 2014. Google Scholar
  19. Yury Polyanskiy, Ananda Theertha Suresh, and Yihong Wu. Sample complexity of population recovery. In Satyen Kale and Ohad Shamir, editors, Proceedings of the 2017 Conference on Learning Theory, volume 65 of Proceedings of Machine Learning Research, pages 1589-1618, Amsterdam, Netherlands, July 2017. PMLR. Google Scholar
  20. Barbara Terhal. Quantum error correction for quantum memories. Reviews of Modern Physics, 87(2):307, 2015. Google Scholar
  21. Martin Wainwright. High-dimensional statistics: A non-asymptotic viewpoint. Cambridge University Press, 2019. Google Scholar
  22. Joel Wallman and Joseph Emerson. Noise tailoring for scalable quantum computation via randomized compiling. Physical Review A, 94(5):052325, 2016. Google Scholar
  23. Avi Wigderson and Amir Yehudayoff. Population recovery and partial identification. Machine Learning, 102(1):29-56, 2016. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail