On the complexity and verification of quantum random circuit sampling


A critical milestone on the path to useful quantum computers is the demonstration of a quantum computation that is prohibitively hard for classical computers—a task referred to as quantum supremacy. A leading near-term candidate is sampling from the probability distributions of randomly chosen quantum circuits, which we call random circuit sampling (RCS). RCS was defined with experimental realizations in mind, leaving its computational hardness unproven. Here we give strong complexity-theoretic evidence of classical hardness of RCS, placing it on par with the best theoretical proposals for supremacy. Specifically, we show that RCS satisfies an average-case hardness condition, which is critical to establishing computational hardness in the presence of experimental noise. In addition, it follows from known results that RCS also satisfies an anti-concentration property, namely that errors in estimating output probabilities are small with respect to the probabilities themselves. This makes RCS the first proposal for quantum supremacy with both of these properties. Finally, we also give a natural condition under which an existing statistical measure, cross-entropy, verifies RCS, as well as describe a new verification measure that in some formal sense maximizes the information gained from experimental samples.

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  1. 1.

    Bernstein, E. & Vazirani, U. V. Quantum complexity theory. In Proc. 25th Annual ACM Symposium on Theory of Computing (eds Kosaraju, S. R. et al.) 11–20 (ACM, 1993).

  2. 2.

    Simon, D. R. On the power of quantum cryptography. In Proc. 35th Annual Symposium on Foundations of Computer Science 116–123 (IEEE Computer Society, 1994).

  3. 3.

    Shor, P. W. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Rev. 41, 303–332 (1999).

    ADS  MathSciNet  Article  Google Scholar 

  4. 4.

    Mohseni, M. et al. Commercialize quantum technologies in five years. Nature 543, 171–174 (2017).

    ADS  Article  Google Scholar 

  5. 5.

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

    ADS  Article  Google Scholar 

  6. 6.

    Zhang, J. et al. Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator. Nature 551, 601–604 (2017).

    ADS  Article  Google Scholar 

  7. 7.

    Preskill, J. Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018).

    Article  Google Scholar 

  8. 8.

    Aaronson, S. & Arkhipov, A. The computational complexity of linear optics. In Proc. 43rd Annual ACM Symposium on Theory of Computing (eds Fortnow, L. & Vadhan, S. P.) 333–342 (ACM, 2011).

  9. 9.

    Boixo, S. et al. Characterizing quantum supremacy in near-term devices. Nat. Phys. 14, 595–600 (2018).

    Article  Google Scholar 

  10. 10.

    Bremner, M. J., Jozsa, R. & Shepherd, D. J. Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy. In Proc. Royal Society of London A: Mathematical, Physical and Engineering Sciences 459–472 (The Royal Society, 2010).

  11. 11.

    Spring, J. B. et al. Boson sampling on a photonic chip. Science 339, 798–801 (2013).

    ADS  Article  Google Scholar 

  12. 12.

    Broome, M. A. et al. Photonic boson sampling in a tunable circuit. Science 339, 794–798 (2013).

    ADS  Article  Google Scholar 

  13. 13.

    Tillmann, M. et al. Experimental boson sampling. Nat. Photonics 7, 540–544 (2013).

    ADS  Article  Google Scholar 

  14. 14.

    Crespi, A. et al. Integrated multimode interferometers with arbitrary designs for photonic boson sampling. Nat. Photonics 7, 545–549 (2013).

    ADS  Article  Google Scholar 

  15. 15.

    Neville, A. et al. No imminent quantum supremacy by boson sampling. Nat. Phys. 13, 1153–1157 (2017).

    Article  Google Scholar 

  16. 16.

    Clifford, P. & Clifford, R. The classical complexity of boson sampling. In Proc. 29th Annual ACM-SIAM Symposium on Discrete Algorithms, (ed. Czumaj, A.) 146–155 (SIAM, 2018).

  17. 17.

    Martinis, J. The quantum space race (2018). Plenary talk at Quantum Information Processing (QIP) 2018. TU Delft https://collegerama.tudelft.nl/Mediasite/Showcase/qip2018/Channel/qip-day3 (2018)

  18. 18.

    Brandão, F. G. & Horodecki, M. Exponential quantum speed-ups are generic. Quantum Inf. Comput. 13, 901–924 (2013).

    MathSciNet  Google Scholar 

  19. 19.

    Hangleiter, D., Bermejo-Vega, J., Schwarz, M. & Eisert, J. Anticoncentration theorems for schemes showing a quantum speedup. Quantum 2, 65 (2018).

    Article  Google Scholar 

  20. 20.

    Terhal, B. M. & DiVincenzo, D. P. Adaptive quantum computation, constant depth quantum circuits and Arthur–Merlin games. Quantum Inf. Comput. 4, 134–145 (2004).

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Morimae, T., Fujii, K. & Fitzsimons, J. F. Hardness of classically simulating the one-clean-qubit model. Phys. Rev. Lett. 112, 130502 (2014).

    ADS  Article  Google Scholar 

  22. 22.

    Farhi, E. & Harrow, A. W. Quantum supremacy through the quantum approximate optimization algorithm. Preprint at https://arxiv.org/abs/1602.07674 (2016).

  23. 23.

    Bouland, A, Mancinska, L. & Zhang, X. Complexity classification of two-qubit commuting Hamiltonians. In Proc. 31st Conference on Computational Complexity (CCC 2016), vol. 50 of Leibniz International Proceedings in Informatics (LIPIcs) (ed. Raz, R.) 28:1–28:33 (Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2016).

  24. 24.

    Lipton, R. J. New directions in testing. In Proc. Distributed Computing and Cryptography, vol. 2 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science (eds Feigenbaum, J. & Merritt, M.) 191–202 (DIMACS/AMS, 1991).

  25. 25.

    Pastawski, F., Yoshida, B., Harlow, D. & Preskill, J. Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence. J. High Energy Phys. 2015, 149 (2015).

    MathSciNet  Article  Google Scholar 

  26. 26.

    Fefferman, B. & Umans, C. On the power of quantum Fourier sampling. In Proc. 11th Conference on the Theory of Quantum Computation, Communication and Cryptography, vol. 61 of Leibniz International Proceedings in Informatics (LIPIcs) (ed. Broadbent, A.) 1:1–1:19 (Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2016).

  27. 27.

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

    ADS  MathSciNet  Article  Google Scholar 

  28. 28.

    Aaronson, S. & Chen, L. Complexity-theoretic foundations of quantum supremacy experiments. In Proc. 32nd Computational Complexity Conference, vol. 79 of LIPIcs (ed. O’Donnell, R.) 22:1–22:67 (Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 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).

    Article  Google Scholar 

  30. 30.

    Morimae, T. Hardness of classically sampling the one-clean-qubit model with constant total variation distance error. Phys. Rev. A 96, 040302 (2017).

    ADS  MathSciNet  Article  Google Scholar 

  31. 31.

    Bouland, A, Fitzsimons, J. F. & Koh, D. E . Complexity classification of conjugated Clifford circuits. In Proc. 33rd Computational Complexity Conference, vol. 102 of Leibniz International Proceedings in Informatics (LIPIcs) (ed. Servedio, R. A.) 21:1–21:25 (Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik: 2018).

  32. 32.

    Mann, R. L. & Bremner, M. J. On the complexity of random quantum computations and the Jones polynomial. Preprint at https://arxiv.org/abs/1711.00686 (2017).

  33. 33.

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

    ADS  MathSciNet  Article  Google Scholar 

  34. 34.

    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).

  35. 35.

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

    ADS  MathSciNet  Article  Google Scholar 

  36. 36.

    Welch, L. & Berlekamp, E. Error correction for algebraic block codes. US patent 4,633,470 (1986).

  37. 37.

    Gemmell, P., Lipton, R., Rubinfeld, R., Sudan, M. & Wigderson, A. Self-testing/correcting for polynomials and for approximate functions. In Proc. 23rd Annual ACM Symposium on Theory of Computing (eds Koutsougeras, C. & Vitter, J. S.) 33–42 (ACM, 1991).

  38. 38.

    Valiant, L. The complexity of computing the permanent. Theor. Comput. Sci. 8, 189–201 (1979).

    MathSciNet  Article  Google Scholar 

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We thank S. Aaronson, D. Aharonov, F. Brandão, M. Coudron, A. Deshpande, T. Gur, Z. Landau, N. Spooner and H. Yuen for helpful discussions. A.B., B.F., C.N. and U.V. were supported by ARO grant W911NF-12-1-0541 and NSF grant CCF-1410022 and a Vannevar Bush faculty fellowship. B.F. is supported in part by an Air Force Office of Scientific Research Young Investigator Program award number FA9550-18-1-0148. Parts of this work were done at the Kavli Institute for Theoretical Physics. Portions of this paper are a contribution of NIST, an agency of the US government, and are not subject to US copyright.

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All authors contributed equally to this work; author ordering is alphabetical.

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Correspondence to Bill Fefferman.

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Bouland, A., Fefferman, B., Nirkhe, C. et al. On the complexity and verification of quantum random circuit sampling. Nature Phys 15, 159–163 (2019). https://doi.org/10.1038/s41567-018-0318-2

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