Pauli Error Estimation via Population Recovery

Authors Steven T. Flammia, Ryan O'Donnell

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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.

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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


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