Skip to main content

Thank you for visiting nature.com. 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.

  • Letter
  • Published:

Error mitigation extends the computational reach of a noisy quantum processor

Nature volume 567, pages 491–495 (2019)Cite this article

Subjects

Abstract

Quantum computation, a paradigm of computing that is completely different from classical methods, benefits from theoretically proved speed-ups for certain problems and can be used to study the properties of quantum systems1. Yet, because of the inherently fragile nature of the physical computing elements (qubits), achieving quantum advantages over classical computation requires extremely low error rates for qubit operations, as well as substantial physical qubits, to realize fault tolerance via quantum error correction2,3. However, recent theoretical work4,5 has shown that the accuracy of computation (based on expectation values of quantum observables) can be enhanced through an extrapolation of results from a collection of experiments of varying noise. Here we demonstrate this error mitigation protocol on a superconducting quantum processor, enhancing its computational capability, with no additional hardware modifications. We apply the protocol to mitigate errors in canonical single- and two-qubit experiments and then extend its application to the variational optimization6,7,8 of Hamiltonians for quantum chemistry and magnetism9. We effectively demonstrate that the suppression of incoherent errors helps to achieve an otherwise inaccessible level of accuracy in the variational solutions using our noisy processor. These results demonstrate that error mitigation techniques will enable substantial improvements in the capabilities of near-term quantum computing hardware.

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

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Device and experimental protocol.
Fig. 2: Error mitigation of random single-qubit and two-qubit circuits.
Fig. 3: Error-mitigated variational optimization of interacting spin Hamiltonians.
Fig. 4: Error-mitigated variational optimization of molecular Hamiltonians.

Similar content being viewed by others

Data availability

The data that support the findings of this study are available from the corresponding author on reasonable request.

References

  1. Feynman, R. P. Simulating physics with computers. Int. J. Theor. Phys. 21, 467–488 (1982).

    Article  MathSciNet  Google Scholar 

  2. Shor, P. W. Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52, 2493–2496 (1995).

    Article  ADS  Google Scholar 

  3. Steane, A. M. Error correcting codes in quantum theory. Phys. Rev. Lett. 77, 793–797 (1996).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  4. Temme, K., Bravyi, S. & Gambetta, J. M. Error mitigation for short-depth quantum circuits. Phys. Rev. Lett. 119, 180509 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  5. Li, Y. & Benjamin, S. C. Efficient variational quantum simulator incorporating active error minimization. Phys. Rev. X 7, 021050 (2017).

    Google Scholar 

  6. Yung, M.-H. et al. From transistor to trapped-ion computers for quantum chemistry. Sci. Rep. 4, 3589 (2014).

    Article  Google Scholar 

  7. Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nat. Commun. 5, 4213 (2014).

    Article  CAS  Google Scholar 

  8. Farhi, E., Goldstone, J. & Gutmann, S. A quantum approximate optimization algorithm. Preprint at https://arxiv.org/abs/1411.4028 (2014).

  9. Kandala, A. et al. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature 549, 242–246 (2017).

    Article  ADS  CAS  Google Scholar 

  10. Preskill, J. Quantum computing in the NISQ era and beyond. Preprint at https://arxiv.org/abs/1801.00862 (2018).

  11. McClean, J., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New J. Phys. 18, 023023 (2016).

    Article  ADS  Google Scholar 

  12. O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016).

    Google Scholar 

  13. Endo, S., Benjamin, S. C. & Li, Y. Practical quantum error mitigation for near-future applications. Phys. Rev. X 8, 031027 (2018).

    CAS  Google Scholar 

  14. Richardson, L. F. IX. The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam. Phil. Trans. R. Soc. Lond. A 210, 307–357 (1911).

    Article  ADS  Google Scholar 

  15. Dumitrescu, E. et al. Cloud quantum computing of an atomic nucleus. Phys. Rev. Lett. 120, 210501 (2018).

  16. Müller, C., Lisenfeld, J., Shnirman, A. & Poletto, S. Interacting two-level defects as sources of fluctuating high-frequency noise in superconducting circuits. Phys. Rev. B 92, 035442 (2015).

    Article  ADS  Google Scholar 

  17. Koch, J. et al. Charge-insensitive qubit design derived from the Cooper pair box. Phys. Rev. A 76, 042319 (2007).

    Article  ADS  Google Scholar 

  18. McKay, D. C., Wood, C. J., Sheldon, S., Chow, J. M. & Gambetta, J. M. Efficient Z gates for quantum computing. Phys. Rev. A 96, 022330 (2017).

    Article  ADS  Google Scholar 

  19. Paraoanu, G. S. Microwave-induced coupling of superconducting qubits. Phys. Rev. B 74, 140504 (2006).

    Article  ADS  Google Scholar 

  20. Sheldon, S., Magesan, E., Chow, J. M. & Gambetta, J. M. Procedure for systematically tuning up cross-talk in the cross-resonance gate. Phys. Rev. A 93, 060302 (2016).

    Article  ADS  Google Scholar 

  21. Magesan, E., Gambetta, J. M. & Emerson, J. Characterizing quantum gates via randomized benchmarking. Phys. Rev. A 85, 042311 (2012).

    Article  ADS  Google Scholar 

  22. Colless, J. I. et al. Computation of molecular spectra on a quantum processor with an error-resilient algorithm. Phys. Rev. X 8, 011021 (2018).

    CAS  Google Scholar 

  23. Hempel, C. et al. Quantum chemistry calculations on a trapped-ion quantum simulator. Phys. Rev. X 8, 031022 (2018).

  24. Spall, J. C. Multivariate stochastic approximation using a simultaneous perturbation gradient approximation. IEEE Trans. Automat. Contr. 37, 332–341 (1992).

    Article  MathSciNet  Google Scholar 

  25. McClean, J. R., Kimchi-Schwartz, M. E., Carter, J. & de Jong, W. A. Hybrid quantum-classical hierarchy for mitigation of decoherence and determination of excited states. Phys. Rev. A 95, 042308 (2017).

    Article  ADS  Google Scholar 

  26. Lanyon, B. P. et al. Towards quantum chemistry on a quantum computer. Nat. Chem. 2, 106–111 (2010).

    Article  CAS  Google Scholar 

  27. Bravyi, S., Gambetta, J. M., Mezzacapo, A. & Temme, K. Tapering off qubits to simulate fermionic Hamiltonians. Preprint at https://arxiv.org/abs/1701.08213 (2017).

  28. Egger, D. J. et al. Pulsed reset protocol for fixed-frequency superconducting qubits. Phys. Rev. X 8, 031022 (2018).

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

    Article  ADS  Google Scholar 

  30. Havlicek, V. et al. Supervised learning with quantum enhanced feature spaces. Preprint at https://arxiv.org/abs/1804.11326 (2018).

  31. Chow, J. M. et al. Implementing a strand of a scalable fault-tolerant quantum computing fabric. Nat. Commun. 5, 4015 (2014).

    Article  CAS  Google Scholar 

  32. 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).

    Article  Google Scholar 

  33. Bergeal, N. et al. Analog information processing at the quantum limit with a josephson ring modulator. Nat. Phys. 6, 296–302 (2010).

    Article  CAS  Google Scholar 

  34. Abdo, B., Schackert, F., Hatridge, M., Rigetti, C. & Devoret, M. Josephson amplifier for qubit readout. Appl. Phys. Lett. 99, 162506 (2011).

    Article  ADS  Google Scholar 

  35. Motzoi, F., Gambetta, J. M., Rebentrost, P. & Wilhelm, F. K. Simple pulses for elimination of leakage in weakly nonlinear qubits. Phys. Rev. Lett. 103, 110501 (2009).

    Article  ADS  CAS  Google Scholar 

  36. Chow, J. M. et al. Simple all-microwave entangling gate for fixed-frequency superconducting qubits. Phys. Rev. Lett. 107, 080502 (2011).

    Article  ADS  Google Scholar 

  37. Córcoles, A. D. et al. Process verification of two-qubit quantum gates by randomized benchmarking. Phys. Rev. A 87, 030301 (2013).

    Article  ADS  Google Scholar 

  38. Rigetti, C. & Devoret, M. Fully microwave-tunable universal gates in superconducting qubits with linear couplings and fixed transition frequencies. Phys. Rev. B 81, 134507 (2010).

    Article  ADS  Google Scholar 

  39. Magesan, E. & Gambetta, J. M. Effective Hamiltonian models of the cross-resonance gate. Preprint at https://arxiv.org/abs/1804.04073 (2018).

  40. Sheldon, S. et al. Characterizing errors on qubit operations via iterative randomized benchmarking. Phys. Rev. A 93, 012301 (2016).

    Article  ADS  Google Scholar 

  41. Efron, B. in Breakthroughs in Statistics 569–593 (Springer, 1992).

Download references

Acknowledgements

We thank J. Rozen, M. Takita, and O. Jinka for experimental contributions, B. Abdo for design and characterization of the Josephson parametric converters, and M. Brink for device fabrication. We acknowledge discussions with S. Brayvi, E. Magesan, S. Sheldon, and M. Takita. We acknowledge support from the IBM Research Frontiers Institute. The research is based on work supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via the Army Research Office contract W911NF-10-1-0324.

Author information

Authors and Affiliations

  1. IBM T. J. Watson Research Center, Yorktown Heights, NY, USA

    Abhinav Kandala, Kristan Temme, Antonio D. Córcoles, Antonio Mezzacapo, Jerry M. Chow & Jay M. Gambetta

Authors
  1. Abhinav Kandala
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kristan Temme
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Antonio D. Córcoles
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Antonio Mezzacapo
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Jerry M. Chow
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Jay M. Gambetta
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

A.K., K.T. and J.M.G. developed the experimental protocol, planned the experiments and analysed the experimental data. A.K. performed the experiments. A.D.C. contributed to the experimental set-up. A.M. contributed to the data analysis. A.K., K.T., J.M.C., and J.M.G. wrote the manuscript, with contributions from all authors.

Corresponding author

Correspondence to Abhinav Kandala.

Ethics declarations

Competing interests

The authors declare that elements of this work are included in a patent filed by the International Business Machines Corporation with the US Patent and Trademark office.

Additional information

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

Extended data figures and tables

Extended Data Fig. 1 Gate characterization.

a, b, Single-qubit (1Q, a) and two-qubit (2Q, b) error per gate (EPG), for a range of stretch factors c, obtained by randomized benchmarking (RB). The error bars represent the standard deviation of the exponential fits to the  RB decay, for each stretch factor. The considered 1Q gates in a are for qubits Q0 (red), Q1 (yellow), Q2 (green), Q3 (blue) and the 2Q gates in b are CR0-1 (red), CR1-2 (green), CR3-2 (yellow).

Extended Data Fig. 2 Coherence and relaxation time fluctuations in superconducting qubits.

a, Repeated measurements of T1 and echo T2 show fluctuations that are larger than the error bars to the fits. The error bars represent the standard deviation of the exponential fits to each instance of the T1 and echo T2 decay curves. b, c, Decay of excited-state projector \(\langle {\Pi }_{1}\rangle \) for a standard T1 sequence (b) and T2 echo sequence (c), for stretch factors c = 1, 2 obtained using 105 samples. In both cases, the average decay over the total sampling duration for the different stretch factors is the same only when their measurements are grouped together (run 1), in contrast to when sampled separately (run 2 and run 3).

Extended Data Fig. 3 Hamiltonian tomography of the CR drive.

Amplitude dependence of the interaction terms in the cross-resonance drive CR3−2. The error bars represent the standard deviation of the interaction strengths, estimated from fits to Hamiltonian tomography20 for each CR drive amplitude. The dominant interactions in the drive are ZX and IX, with discernible nonlinearities in the strong drive limit. (a.u., arbitrary units.).

Extended Data Fig. 4 Error mitigation with a nonlinear CR drive.

Time evolution of \(\left\langle IZ\right\rangle \) under a simplistic CR drive JZX = −0.0159JΩ + 1.0541 × 10−6JΩ3, and fixed amplitude damping and dephasing channels, for drive amplitudes calibrated to ZXπ/2 gate times Tgate values 2J−1 (a), 3J−1 (b) and 6J−1 (c) and with stretch factors c = 1 (blue) and c = 2 (green). The mitigated time evolution (red) goes out of bounds for fast gates (large drive amplitudes), owing to the nonlinearity in the amplitude dependence.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kandala, A., Temme, K., Córcoles, A.D. et al. Error mitigation extends the computational reach of a noisy quantum processor. Nature 567, 491–495 (2019). https://doi.org/10.1038/s41586-019-1040-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41586-019-1040-7

This article is cited by

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Advanced search

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