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Sculpting Quantum Speedups

Authors Scott Aaronson, Shalev Ben-David

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Scott Aaronson
Shalev Ben-David

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Scott Aaronson and Shalev Ben-David. Sculpting Quantum Speedups. In 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 50, pp. 26:1-26:28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.CCC.2016.26

Abstract

Given a problem which is intractable for both quantum and classical algorithms, can we find a sub-problem for which quantum algorithms provide an exponential advantage? We refer to this problem as the "sculpting problem." In this work, we give a full characterization of sculptable functions in the query complexity setting. We show that a total function f can be restricted to a promise P such that Q(f|_P)=O(polylog(N)) and R(f|_P)=N^{Omega(1)}, if and only if f has a large number of inputs with large certificate complexity. The proof uses some interesting techniques: for one direction, we introduce new relationships between randomized and quantum query complexity in various settings, and for the other direction, we use a recent result from communication complexity due to Klartag and Regev. We also characterize sculpting for other query complexity measures, such as R(f) vs. R_0(f) and R_0(f) vs. D(f).

Along the way, we prove some new relationships for quantum query complexity: for example, a nearly quadratic relationship between Q(f) and D(f) whenever the promise of f is small. This contrasts with the recent super-quadratic query complexity separations, showing that the maximum gap between classical and quantum query complexities is indeed quadratic in various settings - just not for total functions!

Lastly, we investigate sculpting in the Turing machine model. We show that if there is any BPP-bi-immune language in BQP, then every language outside BPP can be restricted to a promise which places it in PromiseBQP but not in PromiseBPP. Under a weaker assumption, that some problem in BQP is hard on average for P/poly, we show that every paddable language outside BPP is sculptable in this way.

Subject Classification

Keywords
  • Quantum Computing
  • Query Complexity
  • Decision Tree Complexity
  • Structural Complexity

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References

  1. Scott Aaronson. Quantum certificate complexity. SIAM Journal on Computing, 35(4):804-824, 2006. Google Scholar
  2. Scott Aaronson, Shalev Ben-David, and Robin Kothari. Separations in query complexity using cheat sheets. arXiv preprint http://arxiv.org/abs/arXiv:1511.01937, 2015. Google Scholar
  3. Andris Ambainis, Kaspars Balodis, Aleksandrs Belovs, Troy Lee, Miklos Santha, and Juris Smotrovs. Separations in query complexity based on pointer functions. arXiv preprint http://arxiv.org/abs/arXiv:1506.04719, 2015. Google Scholar
  4. Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, and Ronald de Wolf. Quantum lower bounds by polynomials. Journal of the ACM, 48(4):778-797, 2001. http://arxiv.org/abs/arXiv:quant-ph/9802049, URL: http://dx.doi.org/10.1145/502090.502097.
  5. Leonard Berman and Juris Hartmanis. On isomorphisms and density of np and other complete sets. SIAM Journal on Computing, 6(2):305-322, 1977. Google Scholar
  6. Harry Buhrman and Ronald de Wolf. Complexity measures and decision tree complexity: a survey. Theoretical Computer Science, 288(1):21-43, 2002. URL: http://dx.doi.org/10.1016/S0304-3975(01)00144-X.
  7. Harry Buhrman, Lance Fortnow, Ilan Newman, and Hein Röhrig. Quantum property testing. SIAM Journal on Computing, 37(5):1387-1400, 2008. Google Scholar
  8. Richard Cleve. The query complexity of order-finding. Information and Computation, 192(2):162-171, 2004. Google Scholar
  9. Philippe Flajolet and Jean-Marc Steyaert. On sets having only hard subsets. In Automata, Languages and Programming, pages 446-457. Springer, 1974. Google Scholar
  10. Jorge E Hirsch. An index to quantify an individual’s scientific research output. Proceedings of the National academy of Sciences of the United States of America, 102(46):16569-16572, 2005. Google Scholar
  11. Bo'az Klartag and Oded Regev. Quantum one-way communication can be exponentially stronger than classical communication. In Proceedings of the forty-third annual ACM symposium on Theory of computing, pages 31-40. ACM, 2011. Google Scholar
  12. Ilan Kremer. Quantum communication. PhD thesis, Citeseer, 1995. Google Scholar
  13. Raghav Kulkarni and Avishay Tal. On fractional block sensitivity. Electronic Colloquium on Computational Complexity (ECCC) http://eccc.hpi-web.de/report/2013/168/, 2013.
  14. Stuart A Kurtz, Stephen R Mahaney, and James S Royer. The isomorphism conjecture fails relative to a random oracle. Journal of the ACM (JACM), 42(2):401-420, 1995. Google Scholar
  15. Ran Raz. Exponential separation of quantum and classical communication complexity. In Proceedings of the thirty-first annual ACM symposium on Theory of computing, pages 358-367. ACM, 1999. Google Scholar
  16. Norbert Sauer. On the density of families of sets. Journal of Combinatorial Theory, Series A, 13(1):145-147, 1972. Google Scholar
  17. Saharon Shelah. A combinatorial problem; stability and order for models and theories in infinitary languages. Pacific Journal of Mathematics, 41(1):247-261, 1972. Google Scholar
  18. Peter W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal on Computing, 26(5):1484-1509, 1997. URL: http://arxiv.org/abs/quant-ph/9508027.
  19. Andrew Chi-Chih Yao. Probabilistic computations: toward a unified measure of complexity. In Annual Symposium on Foundations of Computer Science, volume 17, page 222. Institute of Electrical and Electronics Engineers, 1977. Google Scholar
  20. Bohua Zhan, Shelby Kimmel, and Avinatan Hassidim. Super-polynomial quantum speed-ups for boolean evaluation trees with hidden structure. In Proceedings of the 3rd Innovations in Theoretical Computer Science conference, pages 249-265. ACM, 2012. Google Scholar
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