A critical question for quantum computing in the near future is whether quantum devices without error correction can perform a well-defined computational task beyond the capabilities of supercomputers. Such a demonstration of what is referred to as quantum supremacy requires a reliable evaluation of the resources required to solve tasks with classical approaches. Here, we propose the task of sampling from the output distribution of random quantum circuits as a demonstration of quantum supremacy. We extend previous results in computational complexity to argue that this sampling task must take exponential time in a classical computer. We introduce cross-entropy benchmarking to obtain the experimental fidelity of complex multiqubit dynamics. This can be estimated and extrapolated to give a success metric for a quantum supremacy demonstration. We study the computational cost of relevant classical algorithms and conclude that quantum supremacy can be achieved with circuits in a two-dimensional lattice of 7 × 7 qubits and around 40 clock cycles. This requires an error rate of around 0.5% for two-qubit gates (0.05% for one-qubit gates), and it would demonstrate the basic building blocks for a fault-tolerant quantum computer.

Access optionsAccess options

Rent or Buy article

Get time limited or full article access on ReadCube.

from $8.99

All prices are NET prices.

Additional information

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


  1. 1.

    Emerson, J., Weinstein, Y. S., Saraceno, M., Lloyd, S. & Cory, D. G. Pseudo-random unitary operators for quantum information processing. Science 302, 2098–2100 (2003).

  2. 2.

    Scott, A. J., Brun, T. A., Caves, C. M. & Schack, R. Hypersensitivity and chaos signatures in the quantum baker's maps. J. Phys. A 39, 13405–13433 (2006).

  3. 3.

    Oliveira, R., Dahlsten, O. & Plenio, M. Generic entanglement can be generated efficiently. Phys. Rev. Lett. 98, 130502 (2007).

  4. 4.

    Arnaud, L. & Braun, D. Efficiency of producing random unitary matrices with quantum circuits. Phys. Rev. A 78, 062329 (2008).

  5. 5.

    Trail, C. M., Madhok, V. & Deutsch, I. H. Entanglement and the generation of random states in the quantum chaotic dynamics of kicked coupled tops. Phys. Rev. E 78, 046211 (2008).

  6. 6.

    Harrow, A. W. & Low, R. A. Random quantum circuits are approximate 2-designs. Comm. Math. Phys. 291, 257–302 (2009).

  7. 7.

    Weinstein, Y. S., Brown, W. G. & Viola, L. Parameters of pseudo-random quantum circuits. Phys. Rev. A 78, 052332 (2008).

  8. 8.

    Brown, W. & Fawzi, O. Scrambling speed of random quantum circuits. Preprint at https://arxiv.org/abs/1210.6644 (2012).

  9. 9.

    Kim, H. & Huse, D. A. Ballistic spreading of entanglement in a diffusive nonintegrable system. Phys. Rev. Lett. 111, 127205 (2013).

  10. 10.

    Hosur, P., Qi, X.-L., Roberts, D. A. & Yoshida, B. Chaos in quantum channels. J. High. Energy Phys. 2016, 4 (2016).

  11. 11.

    Nahum, A., Ruhman, J., Vijay, S. & Haah, J. Quantum entanglement growth under random unitary dynamics. Phys. Rev. X 7, 031016 (2017).

  12. 12.

    Aaronson, S. Quantum computing, postselection, and probabilistic polynomial-time. Proc. R. Soc. A 461, 3473–3482 (2005).

  13. 13.

    Bremner, M. J., Jozsa, R. & Shepherd, D. J. Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy. Proc. R. Soc. A 467, 459–472 (2011).

  14. 14.

    Aaronson, S. & Arkhipov, A. The computational complexity of linear optics. In STOC '11 Proc. Forty-Third Annual ACM Symp. Theory of Computing 333–342 (ACM, New York, NY, 2011).

  15. 15.

    Fujii, K. & Morimae, T. Commuting quantum circuits and complexity of Ising partition functions. New J. Phys. 19, 033003 (2017).

  16. 16.

    Goldberg, L. A. & Guo, H. The complexity of approximating complex-valued Ising and Tutte partition functions. Comput. Complex. 26, 765–833 (2017).

  17. 17.

    Bremner, M. J., Montanaro, A. & Shepherd, D. J. Average-case complexity versus approximate simulation of commuting quantum computations. Phys. Rev. Lett. 117, 080501 (2016).

  18. 18.

    Preskill, J. Quantum computing and the entanglement frontier. Preprint at https://arxiv.org/abs/1203.5813 (2012).

  19. 19.

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

  20. 20.

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

  21. 21.

    Peres, A. Stability of quantum motion in chaotic and regular systems. Phys. Rev. A 30, 1610 (1984).

  22. 22.

    Schack, R. & Caves, C. M. Hypersensitivity to perturbations in the quantum baker's map. Phys. Rev. Lett. 71, 525 (1993).

  23. 23.

    Gorin, T., Prosen, T., Seligman, T. H. & Žnidarič, M. Dynamics of Loschmidt echoes and fidelity decay. Phys. Rep. 435, 33–156 (2006).

  24. 24.

    Beenakker, C. W. Random-matrix theory of quantum transport. Rev. Mod. Phys. 69, 731 (1997).

  25. 25.

    Mehta, M. L. Random Matrices Vol. 142 (Academic, San Diego, CA, 2004).

  26. 26.

    Porter, C. & Thomas, R. Fluctuations of nuclear reaction widths. Phys. Rev. 104, 483 (1956).

  27. 27.

    Haake, F. Signatures of Quantum Chaos (Springer, Berlin, 1991).

  28. 28.

    Boixo, S., Smelyanskiy, V. N. & Neven, H. Fourier analysis of sampling from noisy chaotic quantum circuits. Preprint at https://arxiv.org/abs/1708.01875 (2017).

  29. 29.

    Bremner, M. J., Montanaro, A. & Shepherd, D. J. Achieving quantum supremacy with sparse and noisy commuting quantum computations. Quantum 1, 8 (2017).

  30. 30.

    Boixo, S., Isakov, S. V., Smelyanskiy, V. N. & Neven, H. Simulation of low-depth quantum circuits as complex undirected graphical models. Preprint at https://arxiv.org/abs/1712.05384 (2017).

  31. 31.

    Markov, I. L. & Shi, Y. Simulating quantum computation by contracting tensor networks. SICOMP 38, 963–981 (2008).

  32. 32.

    Aaronson, S. & Chen, L. Complexity-theoretic foundations of quantum supremacy experiments. CCC’ 17, 22 (2017).

  33. 33.

    Pednault, E. et al. Breaking the 49-qubit barrier in the simulation of quantum circuits. Preprint at https://arxiv.org/abs/1710.05867 (2017).

  34. 34.

    Bremner, M. J., Mora, C. & Winter, A. Are random pure states useful for quantum computation? Phys. Rev. Lett. 102, 190502 (2009).

  35. 35.

    Bravyi, S. & Gosset, D. Improved classical simulation of quantum circuits dominated by Clifford gates. Phys. Rev. Lett. 116, 250501 (2016).

  36. 36.

    Emerson, J., Alicki, R. & Zyczkowski, K. Scalable noise estimation with random unitary operators. J. Opt. B 7, S347–S352 (2005).

  37. 37.

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

  38. 38.

    Magesan, E., Gambetta, J. M. & Emerson, J. Scalable and robust randomized benchmarking of quantum processes. Phys. Rev. Lett. 106, 180504 (2011).

  39. 39.

    Fowler, A. G., Mariantoni, M., Martinis, J. M. & Cleland, A. N. Surface codes: Towards practical large-scale quantum computation. Phys. Rev. A 86, 032324 (2012).

  40. 40.

    Barends, R. et al. Digital quantum simulation of fermionic models with a superconducting circuit. Nat. Commun. 6, 7654 (2015).

  41. 41.

    Boixo, S. & Monras, A. Operational interpretation for global multipartite entanglement. Phys. Rev. Lett. 100, 100503–100504 (2008).

  42. 42.

    Flammia, S. T. & Liu, Y.-K. Direct fidelity estimation from few Pauli measurements. Phys. Rev. Lett. 106, 230501 (2011).

  43. 43.

    Neill, C. et al. A blueprint for demonstrating quantum supremacy with superconducting qubits. Preprint at https://arxiv.org/abs/1709.06678 (2017).

Download references


We especially acknowledge M. Smelyanskiy, from the Parallel Computing Laboratory, Intel Corporation, who performed the simulations of circuits with 6 × 6 and 7 × 6 qubits and wrote the corresponding section in the Supplementary Information. We would like to acknowledge A. Montanaro for multiple suggestions, especially regarding IQP circuits. We would like to thank S. Aaronson, A. Fowler, I. Markov, M. Mohseni and E. Rieffel for discussions. The authors also thank J. Hammond, from the Parallel Computing Laboratory, Intel Corporation, for his useful insights into MPI run-time performance and scalability. This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the US Department of Energy under contract no. DEAC02-05CH11231. M.J.B. has received financial support from the Australian Research Council via the Future Fellowship scheme (project no. FT110101044) and as a member of the ARC Centre of Excellence for Quantum Computation and Communication Technology (project no. CE170100012).

Author information


  1. Google Inc., Venice, CA, USA

    • Sergio Boixo
    • , Vadim N. Smelyanskiy
    • , Ryan Babbush
    • , Nan Ding
    •  & Hartmut Neven
  2. Google Inc., Zurich, Switzerland

    • Sergei V. Isakov
  3. QuAIL, NASA Ames Research Center, Moffett Field, CA, USA

    • Zhang Jiang
  4. SGT Inc., Greenbelt, MD, USA

    • Zhang Jiang
  5. Centre for Quantum Computation and Communication Technology, Centre for Quantum Software and Information, Faculty of Engineering and Information Technology, University of Technology Sydney, Ultimo, New South Wales, Australia

    • Michael J. Bremner
  6. Google Inc., Santa Barbara, CA, USA

    • John M. Martinis
  7. Department of Physics, University of California, Santa Barbara, CA, USA

    • John M. Martinis


  1. Search for Sergio Boixo in:

  2. Search for Sergei V. Isakov in:

  3. Search for Vadim N. Smelyanskiy in:

  4. Search for Ryan Babbush in:

  5. Search for Nan Ding in:

  6. Search for Zhang Jiang in:

  7. Search for Michael J. Bremner in:

  8. Search for John M. Martinis in:

  9. Search for Hartmut Neven in:


S.B. designed the project. S.B. and V.N.S. developed most of the theory. S.V.I. performed numerical studies and designed the specific quantum circuits. All authors contributed to several tasks, such as analysis of theory and results and discussions of the draft.

Competing interests

The authors declare no competing interests.

Corresponding author

Correspondence to Sergio Boixo.

Supplementary information

  1. Supplementary Information

    Supplementary notes, Supplementary Figures, Supplementary references

About this article

Publication history