Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Efficient learning of quantum noise


Noise is the central obstacle to building large-scale quantum computers. Quantum systems with sufficiently uncorrelated and weak noise could be used to solve computational problems that are intractable with current digital computers. There has been substantial progress towards engineering such systems1,2,3,4,5,6,7,8. However, continued progress depends on the ability to characterize quantum noise reliably and efficiently with high precision9. Here, we describe such a protocol and report its experimental implementation on a 14-qubit superconducting quantum architecture. The method returns an estimate of the effective noise and can detect correlations within arbitrary sets of qubits. We show how to construct a quantum noise correlation matrix allowing the easy visualization of correlations between all pairs of qubits, enabling the discovery of long-range two-qubit correlations in the 14-qubit device that had not previously been detected. Our results are the first implementation of a provably rigorous and comprehensive diagnostic protocol capable of being run on state-of-the-art devices and beyond. These results pave the way for noise metrology in next-generation quantum devices, calibration in the presence of crosstalk, bespoke quantum error-correcting codes10 and customized fault-tolerance protocols11 that can greatly reduce the overhead in a quantum computation.

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Fig. 1: Our algorithm for characterizing the entire averaged probability vector.
Fig. 2: Results from the protocol run in single-qubit mode.
Fig. 3: Results from the protocol run in two-qubit mode.
Fig. 4: Illustration of the applicability of the protocol to larger systems.

Data availability

Source data are available for this paper at All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

Code availability

Details of code that was used to analyse the data is available from the corresponding author upon reasonable request.


  1. Schindler, P. et al. Experimental repetitive quantum error correction. Science 332, 1059–1061 (2011).

    ADS  Article  Google Scholar 

  2. Reed, M. D. et al. Realization of three-qubit quantum error correction with superconducting circuits. Nature 482, 382–385 (2012).

    ADS  Article  Google Scholar 

  3. Barends, R. et al. Superconducting quantum circuits at the surface code threshold for fault tolerance. Nature 508, 500–503 (2014).

    ADS  Article  Google Scholar 

  4. Nigg, D. et al. Quantum computations on a topologically encoded qubit. Science 345, 302–305 (2014).

    ADS  MathSciNet  Article  Google Scholar 

  5. Kelly, J. et al. State preservation by repetitive error detection in a superconducting quantum circuit. Nature 519, 66–69 (2015).

    ADS  Article  Google Scholar 

  6. Córcoles, A. D. et al. Demonstration of a quantum error detection code using a square lattice of four superconducting qubits. Nat. Commun. 6, 6979 (2015).

    ADS  Article  Google Scholar 

  7. Ofek, N. et al. Extending the lifetime of a quantum bit with error correction in superconducting circuits. Nature 536, 441–445 (2016).

    ADS  Article  Google Scholar 

  8. Linke, N. M. et al. Fault-tolerant quantum error detection. Sci. Adv. 3, e1701074 (2017).

    ADS  Article  Google Scholar 

  9. Martinis, J. M. Qubit metrology for building a fault-tolerant quantum computer. npj Quantum Inf. 1, 15005 (2015).

    ADS  Article  Google Scholar 

  10. Tuckett, D. K., Bartlett, S. D. & Flammia, S. T. Ultrahigh error threshold for surface codes with biased noise. Phys. Rev. Lett. 120, 050505 (2018).

    ADS  Article  Google Scholar 

  11. Aliferis, P. & Preskill, J. Fault-tolerant quantum computation against biased noise. Phys. Rev. A 78, 052331 (2008).

    ADS  Article  Google Scholar 

  12. Neill, C. et al. A blueprint for demonstrating quantum supremacy with superconducting qubits. Science 360, 195–199 (2018).

    ADS  MathSciNet  Article  Google Scholar 

  13. Chuang, I. L. & Nielsen, M. A. Prescription for experimental determination of the dynamics of a quantum black box. J. Mod. Opt. 44, 2455–2467 (1997).

    ADS  Article  Google Scholar 

  14. Blume-Kohout, R. et al. Demonstration of qubit operations below a rigorous fault tolerance threshold with gate set tomography. Nat. Commun. 8, 14485 (2016).

    Article  Google Scholar 

  15. Kimmel, S., da Silva, M. P., Ryan, C. A., Johnson, B. R. & Ohki, T. Robust extraction of tomographic information via randomized benchmarking. Phys. Rev. X 4, 011050 (2014).

    Google Scholar 

  16. Gross, D., Liu, Y.-K., Flammia, S. T., Becker, S. & Eisert, J. Quantum state tomography via compressed sensing. Phys. Rev. Lett. 105, 150401 (2010).

    ADS  Article  Google Scholar 

  17. Flammia, S. T., Gross, D., Liu, Y.-K. & Eisert, J. Quantum tomography via compressed sensing: error bounds, sample complexity, and efficient estimators. New J. Phys. 14, 095022 (2012).

    ADS  Article  Google Scholar 

  18. Riofrío, C. A. et al. Experimental quantum compressed sensing for a seven-qubit system. Nat. Commun. 8, 15305 (2017).

    ADS  Article  Google Scholar 

  19. Emerson, J., Alicki, R. & Życzkowski, K. Scalable noise estimation with random unitary operators. J. Opt. B 7, S347 (2005).

    ADS  MathSciNet  Article  Google Scholar 

  20. Knill, E. et al. Randomized benchmarking of quantum gates. Phys. Rev. A 77, 012307 (2008).

    ADS  Article  Google Scholar 

  21. França, D. S. & Hashagen, A. K. Approximate randomized benchmarking for finite groups. J. Phys. A 51, 395302 (2018).

    MathSciNet  Article  Google Scholar 

  22. Proctor, T. J. et al. Direct randomized benchmarking for multiqubit devices. Phys. Rev. Lett. 123, 030503 (2019).

    ADS  Article  Google Scholar 

  23. Wallman, J., Granade, C., Harper, R. & Flammia, S. T. Estimating the coherence of noise. New J. Phys. 17, 113020 (2015).

    ADS  Article  Google Scholar 

  24. Helsen, J., Xue, X., Vandersypen, L. M. K. & Wehner, S. A new class of efficient randomized benchmarking protocols. npj Quantum Inf. 5, 71 (2019).

    ADS  Article  Google Scholar 

  25. Erhard, A. et al. Characterizing large-scale quantum computers via cycle benchmarking. Nat. Commun. 10, 5347 (2019).

    ADS  Article  Google Scholar 

  26. Emerson, J. et al. Symmetrized characterization of noisy quantum processes. Science 317, 1893–1896 (2007).

    ADS  Article  Google Scholar 

  27. Gambetta, J. M. et al. Characterization of addressability by simultaneous randomized benchmarking. Phys. Rev. Lett. 109, 240504 (2012).

    ADS  Article  Google Scholar 

  28. Flammia, S. T. & Wallman, J. J. Efficient estimation of Pauli channels. Preprint at (2019).

  29. Knill, E. Quantum computing with realistically noisy devices. Nature 434, 39–44 (2005).

    ADS  Article  Google Scholar 

  30. Wallman, J. J. & Emerson, J. Noise tailoring for scalable quantum computation via randomized compiling. Phys. Rev. A 94, 052325 (2016).

    ADS  Article  Google Scholar 

  31. Ware, M. et al. Experimental demonstration of Pauli-frame randomization on a superconducting qubit. Preprint at (2018).

  32. Wright, K. et al. Benchmarking an 11-qubit quantum computer. Nat. Commun. 10, 5464 (2019).

    ADS  Article  Google Scholar 

  33. Chubb, C. T. & Flammia, S. T. Statistical mechanical models for quantum codes with correlated noise. Preprint at (2018).

  34. Koller, D. & Friedman, N. Probabilistic Graphical Models: Principles and Techniques (MIT Press, 2009).

  35. Aleksandrowicz, G. et al. Qiskit: an open-source framework for quantum computing. Zenodo (2019).

  36. Harper, R. & Flammia, S. T. Fault-tolerant logical gates in the IBM quantum experience. Phys. Rev. Lett. 122, 080504 (2019).

    ADS  Article  Google Scholar 

  37. Harper, R., Hincks, I., Ferrie, C., Flammia, S. T. & Wallman, J. J. Statistical analysis of randomized benchmarking. Phys. Rev. A 99, 052350 (2019).

    ADS  Article  Google Scholar 

Download references


We thank S. Bartlett, A. Doherty, J. Emerson and T. Monz for comments on an earlier draft. This work was supported in part by US Army Research Office grant nos. W911NF-14-1-0098 and W911NF-14-1-0103, the Australian Research Council Centre of Excellence for Engineered Quantum Systems (EQUS) CE170100009, the Government of Ontario, and the Government of Canada through the Canada First Research Excellence Fund (CFREF) and Transformative Quantum Technologies (TQT), Natural Sciences and Engineering Research Council (NSERC), Industry Canada.

Author information

Authors and Affiliations



R.H., S.T.F. and J.J.W. conceived the experiments, and S.T.F. and J.J.W. conceived the original methodology. The implementation was carried out by R.H. R.H. wrote the initial draft and all authors contributed to the revisions and editing of the manuscript.

Corresponding author

Correspondence to Steven T. Flammia.

Ethics declarations

Competing interests

J.J.W. is the chief technology officer of the company Quantum Benchmark, Inc., and S.T.F. and R.H. were both consultants for it for part of the duration of this project.

Additional information

Peer review information Nature Physics thanks Jonas Bylander, Diego Risté, Marcus da Silva and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Figs. 1–6, discussion and Tables 1–7.

Source data

Source Data Fig. 2

Correlation matrix data, high and low error bounds.

Source Data Fig. 3

Correlation matrix data, high and low error bounds.

Source Data Fig. 4

Data used for Fig. 4 plot.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Harper, R., Flammia, S.T. & Wallman, J.J. Efficient learning of quantum noise. Nat. Phys. 16, 1184–1188 (2020).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

Further reading


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing