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Complexity-Theoretic Foundations of Quantum Supremacy Experiments

Authors Scott Aaronson, Lijie Chen

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Scott Aaronson
Lijie Chen

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Scott Aaronson and Lijie Chen. Complexity-Theoretic Foundations of Quantum Supremacy Experiments. In 32nd Computational Complexity Conference (CCC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 79, pp. 22:1-22:67, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.CCC.2017.22

Abstract

In the near future, there will likely be special-purpose quantum computers with 40-50 high-quality qubits. This paper lays general theoretical foundations for how to use such devices to demonstrate "quantum supremacy": that is, a clear quantum speedup for some task, motivated by the goal of overturning the Extended Church-Turing Thesis as confidently as possible. First, we study the hardness of sampling the output distribution of a random quantum circuit, along the lines of a recent proposal by by the Quantum AI group at Google. We show that there's a natural average-case hardness assumption, which has nothing to do with sampling, yet implies that no polynomial-time classical algorithm can pass a statistical test that the quantum sampling procedure's outputs do pass. Compared to previous work - for example, on BosonSampling and IQP - the central advantage is that we can now talk directly about the observed outputs, rather than about the distribution being sampled. Second, in an attempt to refute our hardness assumption, we give a new algorithm, inspired by Savitch's Theorem, for simulating a general quantum circuit with n qubits and m gates in polynomial space and m^O(n) time. We then discuss why this and other known algorithms fail to refute our assumption. Third, resolving an open problem of Aaronson and Arkhipov, we show that any strong quantum supremacy theorem - of the form "if approximate quantum sampling is classically easy, then the polynomial hierarchy collapses" - must be non-relativizing. This sharply contrasts with the situation for exact sampling. Fourth, refuting a conjecture by Aaronson and Ambainis, we show that the Fourier Sampling problem achieves a constant versus linear separation between quantum and randomized query complexities. Fifth, in search of a "happy medium" between black-box and non-black-box arguments, we study quantum supremacy relative to oracles in P/poly. Previous work implies that, if one-way functions exist, then quantum supremacy is possible relative to such oracles. We show, conversely, that some computational assumption is needed: if SampBPP=SampBQP and NP is in BPP, then quantum supremacy is impossible relative to oracles with small circuits.
Keywords
  • computational complexity
  • quantum computing
  • quantum supremacy

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References

  1. S. Aaronson. BQP and the polynomial hierarchy. In Proc. ACM STOC, 2010. arXiv:0910.4698. Google Scholar
  2. S. Aaronson. Google, D-wave, and the case of the factor-10^8 speedup for WHAT?, 2015. URL: http://www.scottaaronson.com/blog/?p=2555.
  3. S. Aaronson and A. Arkhipov. The computational complexity of linear optics. Theory of Computing, 9(4):143-252, 2013. Earlier version in Proc. ACM STOC'2011. ECCC TR10-170, arXiv:1011.3245. Google Scholar
  4. S. Aaronson et al. The Complexity Zoo. URL: http://www.complexityzoo.com.
  5. S. Aaronson and Y. Shi. Quantum lower bounds for the collision and the element distinctness problems. J. of the ACM, 51(4):595-605, 2004. Google Scholar
  6. S. Aaronson and A. Wigderson. Algebrization: a new barrier in complexity theory. ACM Trans. on Computation Theory, 1(1), 2009. Earlier version in Proc. ACM STOC'2008. Google Scholar
  7. Sccot Aaronson. New evidence that quantum mechanics is hard to simulate on classical computers. http://www.scottaaronson.com/talks/newev.ppt, 2009.
  8. Scott Aaronson. The equivalence of sampling and searching. Theory of Computing Systems, 55(2):281-298, 2014. Google Scholar
  9. Scott Aaronson and Andris Ambainis. Forrelation: A problem that optimally separates quantum from classical computing. In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, pages 307-316. ACM, 2015. Google Scholar
  10. Scott Aaronson, Shalev Ben-David, and Robin Kothari. Separations in query complexity using cheat sheets. arXiv preprint arXiv:1511.01937, 2015. Google Scholar
  11. Scott Aaronson, Adam Bouland, Greg Kuperberg, and Saeed Mehraban. The computational complexity of ball permutations. arXiv preprint arXiv:1610.06646, 2016. Google Scholar
  12. D. Aharonov and M. Ben-Or. Fault-tolerant quantum computation with constant error. In Proc. ACM STOC, pages 176-188, 1997. quant-ph/9906129. Google Scholar
  13. D. Aharonov, M. Ben-Or, and E. Eban. Interactive proofs for quantum computations. arXiv:0810.5375, 2008. Google Scholar
  14. Andris Ambainis. Polynomial degree and lower bounds in quantum complexity: Collision and element distinctness with small range. Theory of Computing, 1(1):37-46, 2005. Google Scholar
  15. Sanjeev Arora and Boaz Barak. Computational complexity: a modern approach. Cambridge University Press, 2009. Google Scholar
  16. Theodore Baker, John Gill, and Robert Solovay. Relativizations of the P=?NP question. SIAM Journal on computing, 4(4):431-442, 1975. Google Scholar
  17. C. Bennett, E. Bernstein, G. Brassard, and U. Vazirani. Strengths and weaknesses of quantum computing. SIAM J. Comput., 26(5):1510-1523, 1997. quant-ph/9701001. Google Scholar
  18. E. Bernstein and U. Vazirani. Quantum complexity theory. SIAM J. Comput., 26(5):1411-1473, 1997. Earlier version in Proc. ACM STOC'1993. Google Scholar
  19. Sergio Boixo, Sergei V Isakov, Vadim N Smelyanskiy, Ryan Babbush, Nan Ding, Zhang Jiang, John M Martinis, and Hartmut Neven. Characterizing quantum supremacy in near-term devices. arXiv preprint arXiv:1608.00263, 2016. Google Scholar
  20. Dan Boneh and Richard J. Lipton. Quantum cryptanalysis of hidden linear functions. In Annual International Cryptology Conference, pages 424-437. Springer, 1995. Google Scholar
  21. Fernando G. S. L. Brandão, Aram W. Harrow, and Michał Horodecki. Local random quantum circuits are approximate polynomial-designs. Communications in Mathematical Physics, 346(2):397-434, 2016. Google Scholar
  22. Sergey Bravyi and David Gosset. Improved classical simulation of quantum circuits dominated by clifford gates. arXiv preprint arXiv:1601.07601, 2016. Google Scholar
  23. M. Bremner, R. Jozsa, and D. Shepherd. Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy. Proc. Roy. Soc. London, A467(2126):459-472, 2010. arXiv:1005.1407. Google Scholar
  24. Michael J. Bremner, Ashley Montanaro, and Dan J. Shepherd. Average-case complexity versus approximate simulation of commuting quantum computations. arXiv preprint arXiv:1504.07999, 2015. Google Scholar
  25. Michael J. Bremner, Ashley Montanaro, and Dan J. Shepherd. Achieving quantum supremacy with sparse and noisy commuting quantum computations. arXiv preprint arXiv:1610.01808, 2016. Google Scholar
  26. A. Broadbent, J. Fitzsimons, and E. Kashefi. Universal blind quantum computation. In Proc. IEEE FOCS, 2009. arXiv:0807.4154. Google Scholar
  27. Jacques Carolan, Christopher Harrold, Chris Sparrow, Enrique Martín-López, Nicholas J Russell, Joshua W Silverstone, Peter J Shadbolt, Nobuyuki Matsuda, Manabu Oguma, Mikitaka Itoh, Graham D Marshall, Mark G Thompson, Jonathan C F Matthews, Toshikazu Hashimoto, Jeremy L O'Brien, and Anthony Laing. Universal linear optics. Science, 349(6249):711-716, 2015. Google Scholar
  28. Lijie Chen. A note on oracle separations for BQP. arXiv preprint arXiv:1605.00619, 2016. Google Scholar
  29. Edward Farhi and Aram W. Harrow. Quantum supremacy through the quantum approximate optimization algorithm. arXiv preprint arXiv:1602.07674, 2016. Google Scholar
  30. L. Fortnow and J. Rogers. Complexity limitations on quantum computation. J. Comput. Sys. Sci., 59(2):240-252, 1999. cs.CC/9811023. Google Scholar
  31. Keisuke Fujii. Noise threshold of quantum supremacy. arXiv preprint arXiv:1610.03632, 2016. Google Scholar
  32. O. Goldreich, S. Goldwasser, and S. Micali. How to construct random functions. J. of the ACM, 33(4):792-807, 1986. Earlier version in Proc. IEEE FOCS'1984, pp. 464-479. Google Scholar
  33. Oded Goldreich and Leonid A Levin. A hard-core predicate for all one-way functions. In Proceedings of the twenty-first annual ACM symposium on Theory of computing, pages 25-32. ACM, 1989. Google Scholar
  34. J. Håstad, R. Impagliazzo, L. A. Levin, and M. Luby. A pseudorandom generator from any one-way function. SIAM J. Comput., 28(4):1364-1396, 1999. Google Scholar
  35. Johan Hastad. Almost optimal lower bounds for small depth circuits. In Proceedings of the eighteenth annual ACM symposium on Theory of computing, pages 6-20. ACM, 1986. Google Scholar
  36. R. Impagliazzo and A. Wigderson. P=BPP unless E has subexponential circuits: derandomizing the XOR Lemma. In Proc. ACM STOC, pages 220-229, 1997. Google Scholar
  37. Richard Jozsa and Marrten Van den Nest. Classical simulation complexity of extended clifford circuits. Quantum Information &Computation, 14(7&8):633-648, 2014. Google Scholar
  38. Gil Kalai. How quantum computers fail: quantum codes, correlations in physical systems, and noise accumulation. arXiv preprint arXiv:1106.0485, 2011. Google Scholar
  39. J. Kelly, R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Yu Chen, et al. State preservation by repetitive error detection in a superconducting quantum circuit. Nature, 519(7541):66-69, 2015. Google Scholar
  40. Samuel Kutin. Quantum lower bound for the collision problem with small range. Theory of Computing, 1(1):29-36, 2005. Google Scholar
  41. Leonid A. Levin. The tale of one-way functions. Problems of Information Transmission, 39(1):92-103, 2003. Google Scholar
  42. Michael Luby and Charles Rackoff. How to construct pseudorandom permutations from pseudorandom functions. SIAM Journal on Computing, 17(2):373-386, 1988. Google Scholar
  43. Igor L. Markov and Yaoyun Shi. Simulating quantum computation by contracting tensor networks. SIAM Journal on Computing, 38(3):963-981, 2008. Google Scholar
  44. Tomoyuki Morimae, Keisuke Fujii, and Joseph F Fitzsimons. Hardness of classically simulating the one-clean-qubit model. Physical review letters, 112(13):130502, 2014. Google Scholar
  45. Noam Nisan and Avi Wigderson. Hardness vs randomness. Journal of computer and System Sciences, 49(2):149-167, 1994. Google Scholar
  46. Borja Peropadre, Gian Giacomo Guerreschi, Joonsuk Huh, and Alán Aspuru-Guzik. Microwave boson sampling. arXiv preprint arXiv:1510.08064, 2015. Google Scholar
  47. John Preskill. Quantum computing and the entanglement frontier. arXiv preprint arXiv:1203.5813, 2012. Google Scholar
  48. A. A. Razborov and S. Rudich. Natural proofs. J. Comput. Sys. Sci., 55(1):24-35, 1997. Earlier version in Proc. ACM STOC'1994, pp. 204-213. Google Scholar
  49. Benjamin Rossman, Rocco A Servedio, and Li-Yang Tan. An average-case depth hierarchy theorem for boolean circuits. In Foundations of Computer Science (FOCS), 2015 IEEE 56th Annual Symposium on, pages 1030-1048. IEEE, 2015. Google Scholar
  50. Terry Rudolph. Why I am optimistic about the silicon-photonic route to quantum computing. arXiv preprint arXiv:1607.08535, 2016. Google Scholar
  51. Amit Sahai and Salil Vadhan. A complete problem for statistical zero knowledge. Journal of the ACM (JACM), 50(2):196-249, 2003. Google Scholar
  52. W. J. Savitch. Relationships between nondeterministic and deterministic tape complexities. J. Comput. Sys. Sci., 4(2):177-192, 1970. Google Scholar
  53. Rocco A Servedio and Steven J Gortler. Equivalences and separations between quantum and classical learnability. SIAM Journal on Computing, 33(5):1067-1092, 2004. Google Scholar
  54. A. Shamir. IP=PSPACE. J. of the ACM, 39(4):869-877, 1992. Earlier version in Proc. IEEE FOCS'1990, pp. 11-15. Google Scholar
  55. P. W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput., 26(5):1484-1509, 1997. Earlier version in Proc. IEEE FOCS'1994. quant-ph/9508027. Google Scholar
  56. Joel Spencer. Asymptopia, volume 71. American Mathematical Soc., 2014. Google Scholar
  57. B. M. Terhal and D. P. DiVincenzo. Adaptive quantum computation, constant-depth circuits and Arthur-Merlin games. Quantum Information and Computation, 4(2):134-145, 2004. quant-ph/0205133. Google Scholar
  58. S. Toda. PP is as hard as the polynomial-time hierarchy. SIAM J. Comput., 20(5):865-877, 1991. Earlier version in Proc. IEEE FOCS'1989, pp. 514-519. Google Scholar
  59. Vladimir Naumovich Vapnik. Statistical learning theory, volume 1. Wiley New York, 1998. Google Scholar
  60. Andrew Chi-Chih Yao. Separating the polynomial-time hierarchy by oracles. In 26th Annual Symposium on Foundations of Computer Science (sfcs 1985), 1985. Google Scholar
  61. Mark Zhandry. How to construct quantum random functions. In Foundations of Computer Science (FOCS), 2012 IEEE 53rd Annual Symposium on, pages 679-687. IEEE, 2012. Google Scholar
  62. Mark Zhandry. A note on quantum-secure prps. arXiv preprint arXiv:1611.05564, 2016. Google Scholar
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