BQP, Meet NP: Search-To-Decision Reductions and Approximate Counting

Authors Sevag Gharibian, Jonas Kamminga



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Author Details

Sevag Gharibian
  • Department of Computer Science and Institute for Photonic Quantum Systems (PhoQS), Paderborn University, Germany
Jonas Kamminga
  • Department of Computer Science and Institute for Photonic Quantum Systems (PhoQS), Paderborn University, Germany

Acknowledgements

The authors would like to thank Ronald de Wolf, Dieter van Melkebeek, Osamu Watanabe, Henry Yuen, Scott Aaronson, William Kretschmer and Eric Allender for helpful comments and remarks.

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Sevag Gharibian and Jonas Kamminga. BQP, Meet NP: Search-To-Decision Reductions and Approximate Counting. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 70:1-70:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.70

Abstract

What is the power of polynomial-time quantum computation with access to an NP oracle? In this work, we focus on two fundamental tasks from the study of Boolean satisfiability (SAT) problems: search-to-decision reductions, and approximate counting. We first show that, in strong contrast to the classical setting where a poly-time Turing machine requires Θ(n) queries to an NP oracle to compute a witness to a given SAT formula, quantumly Θ(log n) queries suffice. We then show this is tight in the black-box model - any quantum algorithm with "NP-like" query access to a formula requires Ω(log n) queries to extract a solution with constant probability. Moving to approximate counting of SAT solutions, by exploiting a quantum link between search-to-decision reductions and approximate counting, we show that existing classical approximate counting algorithms are likely optimal. First, we give a lower bound in the "NP-like" black-box query setting: Approximate counting requires Ω(log n) queries, even on a quantum computer. We then give a "white-box" lower bound (i.e. where the input formula is not hidden in the oracle) - if there exists a randomized poly-time classical or quantum algorithm for approximate counting making o(log n) NP queries, then BPP^NP[o(n)] contains a 𝖯^NP-complete problem if the algorithm is classical and FBQP^NP[o(n)] contains an FP^NP-complete problem if the algorithm is quantum.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
Keywords
  • Approximate Counting
  • Search to Decision Reduction
  • BQP
  • NP
  • Oracle Complexity Class

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References

  1. Scott Aaronson. The equivalence of sampling and searching. Theory of Computing Systems, 55(2):281-298, 2014. URL: https://doi.org/10.1007/s00224-013-9527-3.
  2. Scott Aaronson and Alex Arkhipov. The computational complexity of linear optics. In Proceedings of the forty-third annual ACM symposium on Theory of computing, pages 333-342, 2011. URL: https://doi.org/10.1145/1993636.1993682.
  3. Scott Aaronson, Harry Buhrman, and William Kretschmer. A qubit, a coin, and an advice string walk into a relational problem. arXiv preprint arXiv:2302.10332, 2023. URL: https://doi.org/10.48550/arXiv.2302.10332.
  4. Scott Aaronson, DeVon Ingram, and William Kretschmer. The Acrobatics of BQP. In Shachar Lovett, editor, 37th Computational Complexity Conference (CCC 2022), volume 234 of Leibniz International Proceedings in Informatics (LIPIcs), pages 20:1-20:17, Dagstuhl, Germany, 2022. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.CCC.2022.20.
  5. Andris Ambainis. A better lower bound for quantum algorithms searching an ordered list. In 40th Annual Symposium on Foundations of Computer Science (Cat. No. 99CB37039), pages 352-357. IEEE, 1999. URL: https://doi.org/10.1109/SFFCS.1999.814606.
  6. Andris Ambainis. Quantum lower bounds by quantum arguments. In Proceedings of the thirty-second annual ACM symposium on Theory of computing, pages 636-643, 2000. URL: https://doi.org/10.1145/335305.335394.
  7. Sanjeev Arora and Boaz Barak. Computational complexity: a modern approach. Cambridge University Press, 2009. URL: http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521424264.
  8. Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, and Ronald de Wolf. Quantum lower bounds by polynomials. Journal of the ACM (JACM), 48(4):778-797, 2001. URL: https://doi.org/10.1145/502090.502097.
  9. Shai Ben-David, Benny Chor, and Oded Goldreich. On the theory of average case complexity. In Proceedings of the twenty-first annual ACM symposium on Theory of computing, pages 204-216, 1989. URL: https://doi.org/10.1145/73007.73027.
  10. Charles H Bennett, Ethan Bernstein, Gilles Brassard, and Umesh Vazirani. Strengths and weaknesses of quantum computing. SIAM journal on Computing, 26(5):1510-1523, 1997. URL: https://doi.org/10.1137/S0097539796300933.
  11. Ethan Bernstein and Umesh Vazirani. Quantum complexity theory. In Proceedings of the twenty-fifth annual ACM symposium on Theory of computing, pages 11-20, 1993. URL: https://doi.org/10.1145/167088.167097.
  12. Harry Buhrman and Wim van Dam. Quantum bounded query complexity. In Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference)(Cat. No. 99CB36317), pages 149-156. IEEE, 1999. URL: https://doi.org/10.1109/CCC.1999.766273.
  13. Supratik Chakraborty, Kuldeep S Meel, and Moshe Y Vardi. Algorithmic improvements in approximate counting for probabilistic inference: from linear to logarithmic SAT calls. In Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence, pages 3569-3576, 2016. URL: http://www.ijcai.org/Abstract/16/503.
  14. Remi Delannoy and Kuldeep S Meel. On almost-uniform generation of SAT solutions: The power of 3-wise independent hashing. In Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science, pages 1-10, 2022. URL: https://doi.org/10.1145/3531130.3533338.
  15. Holger Dell, Valentine Kabanets, Dieter van Melkebeek, and Osamu Watanabe. Is Valiant-Vazirani’s isolation probability improvable? computational complexity, 22:345-383, 2013. URL: https://doi.org/10.1007/s00037-013-0059-7.
  16. Lov K Grover. A fast quantum mechanical algorithm for database search. In Proceedings of the twenty-eighth annual ACM symposium on Theory of computing, pages 212-219, 1996. URL: https://doi.org/10.1145/237814.237866.
  17. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-SAT. Journal of Computer and System Sciences, 62(2):367-375, 2001. URL: https://doi.org/10.1006/jcss.2000.1727.
  18. Sandy Irani, Anand Natarajan, Chinmay Nirkhe, Sujit Rao, and Henry Yuen. Quantum Search-To-Decision Reductions and the State Synthesis Problem. In Shachar Lovett, editor, 37th Computational Complexity Conference (CCC 2022), volume 234 of Leibniz International Proceedings in Informatics (LIPIcs), pages 5:1-5:19, Dagstuhl, Germany, 2022. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.CCC.2022.5.
  19. Akinori Kawachi, Benjamin Rossman, and Osamu Watanabe. Query complexity and error tolerance of witness finding algorithms. In Electron. Colloquium Comput. Complex., volume 19, page 2, 2012. URL: https://eccc.weizmann.ac.il/report/2012/002, URL: https://arxiv.org/abs/TR12-002.
  20. Akinori Kawachi, Benjamin Rossman, and Osamu Watanabe. The query complexity of witness finding. Theory of Computing Systems, 61:305-321, 2017. URL: https://doi.org/10.1007/s00224-016-9708-y.
  21. Mark W Krentel. The complexity of optimization problems. In Proceedings of the eighteenth annual ACM symposium on Theory of computing, pages 69-76, 1986. URL: https://doi.org/10.1145/12130.12138.
  22. Larry Stockmeyer. The complexity of approximate counting. In Proceedings of the fifteenth annual ACM symposium on Theory of computing, pages 118-126, 1983. URL: https://doi.org/10.1145/800061.808740.
  23. Leslie G Valiant and Vijay V Vazirani. NP is as easy as detecting unique solutions. In Proceedings of the seventeenth annual ACM symposium on Theory of computing, pages 458-463, 1985. URL: https://doi.org/10.1145/22145.22196.