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Characterizing quantum supremacy in near-term devices


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.

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Fig. 1: Sensitivity to errors of the output distribution of chaotic circuits.
Fig. 2: Convergence to the Porter–Thomas (or exponential) distribution.
Fig. 3: Numerical upper bound for the treewidth.
Fig. 4: Cross-entropy benchmarking and fidelity.

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  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).

    Article  ADS  MathSciNet  Google Scholar 

  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).

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  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).

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

  8. Brown, W. & Fawzi, O. Scrambling speed of random quantum circuits. Preprint at (2012).

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

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  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).

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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).

    Article  ADS  MathSciNet  Google Scholar 

  18. Preskill, J. Quantum computing and the entanglement frontier. Preprint at (2012).

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

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

    Article  ADS  Google Scholar 

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

    Book  Google Scholar 

  28. Boixo, S., Smelyanskiy, V. N. & Neven, H. Fourier analysis of sampling from noisy chaotic quantum circuits. Preprint at (2017).

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

    Article  Google Scholar 

  30. Boixo, S., Isakov, S. V., Smelyanskiy, V. N. & Neven, H. Simulation of low-depth quantum circuits as complex undirected graphical models. Preprint at (2017).

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

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  33. Pednault, E. et al. Breaking the 49-qubit barrier in the simulation of quantum circuits. Preprint at (2017).

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  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).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  43. Neill, C. et al. A blueprint for demonstrating quantum supremacy with superconducting qubits. Preprint at (2017).

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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).

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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.

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Correspondence to Sergio Boixo.

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Boixo, S., Isakov, S.V., Smelyanskiy, V.N. et al. Characterizing quantum supremacy in near-term devices. Nature Phys 14, 595–600 (2018).

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