Abstract
We prove a simple, nearly tight lower bound on the approximate degree of the two-level AND-OR tree using symmetrization arguments. Specifically, we show that ˜deg(ANDm ˆ ORn) = ˜Ω(√mn). We prove this lower bound via reduction to the OR function through a series of symmetrization steps, in contrast to most other proofs that involve formulating approximate degree as a linear program [6, 10, 21]. Our proof also demonstrates the power of a symmetrization technique involving Laurent polynomials (polynomials with negative exponents) that was previously introduced by Aaronson et al. [2].
- Scott Aaronson. 2008. The polynomial method in quantum and classical computing. In 49th Annual IEEE Symposium on Foundations of Computer Science. 3--3. DOI:https://doi.org/10.1109/FOCS.2008.91 Google Scholar
Digital Library
- Scott Aaronson, Robin Kothari, William Kretschmer, and Justin Thaler. 2020. Quantum lower bounds for approximate counting via Laurent polynomials. In 35th Computational Complexity Conference (CCC’20) (Leibniz International Proceedings in Informatics (LIPIcs)), Shubhangi Saraf (Ed.), Vol. 169. Schloss Dagstuhl--Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 7:1--7:47. DOI:https://doi.org/10.4230/LIPIcs.CCC.2020.7 Google Scholar
Digital Library
- Andris Ambainis. 2000. Quantum lower bounds by quantum arguments. In Proceedings of the 32nd Annual ACM Symposium on Theory of Computing (STOC’00). ACM, New York, NY, 636--643. DOI:https://doi.org/10.1145/335305.335394 Google Scholar
Digital Library
- Andris Ambainis. 2005. Polynomial degree and lower bounds in quantum complexity: Collision and element distinctness with small range. Theory of Computing 1, 3 (2005), 37--46. DOI:https://doi.org/10.4086/toc.2005.v001a003Google Scholar
Cross Ref
- Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, and Ronald de Wolf. 2001. Quantum lower bounds by polynomials. Journal of the ACM 48, 4 (July 2001), 778--797. DOI:https://doi.org/10.1145/502090.502097 Google Scholar
Digital Library
- Shalev Ben-David, Adam Bouland, Ankit Garg, and Robin Kothari. 2018. Classical lower bounds from quantum upper bounds. In Proceedings of the 2018 59th Annual IEEE Symposium on Foundations of Computer Science (FOCS’18). https://doi.org/10.1109/FOCS.2009.18Google Scholar
Cross Ref
- Harry Buhrman, Richard Cleve, and Avi Wigderson. 1998. Quantum vs. classical communication and computation. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing (STOC’98). ACM, New York, NY, 63--68. DOI:https://doi.org/10.1145/276698.276713 Google Scholar
Digital Library
- Harry Buhrman and Ronald de Wolf. 2002. Complexity measures and decision tree complexity: a survey. Theoretical Computer Science 288, 1 (2002), 21--43. DOI:https://doi.org/10.1016/S0304-3975(01)00144-X Complexity and Logic. Google Scholar
Digital Library
- Mark Bun, Robin Kothari, and Justin Thaler. 2018. The polynomial method strikes back: Tight quantum query bounds via dual polynomials. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing (STOC’18). Association for Computing Machinery, New York, NY, 297--310. DOI:https://doi.org/10.1145/3188745.3188784 Google Scholar
Digital Library
- Mark Bun and Justin Thaler. 2013. Dual lower bounds for approximate degree and Markov-Bernstein inequalities. In Proceedings of the 40th International Conference on Automata, Languages, and Programming - Volume Part I (ICALP’13). Springer-Verlag, Berlin, 303--314. DOI:https://doi.org/10.1007/978-3-642-39206-1_26 Google Scholar
Digital Library
- Mark Bun and Justin Thaler. 2016. Dual polynomials for collision and element distinctness. Theory of Computing 12, 16 (2016), 1--34. DOI:https://doi.org/10.4086/toc.2016.v012a016Google Scholar
Cross Ref
- Lov K. Grover. 1996. A fast quantum mechanical algorithm for database search. In Proceedings of the 28th Annual ACM Symposium on Theory of Computing (STOC’96). ACM, New York, NY, 212--219. DOI:https://doi.org/10.1145/237814.237866 Google Scholar
Digital Library
- Peter Høyer, Michele Mosca, and Ronald de Wolf. 2003. Quantum search on bounded-error inputs. In Proceedings of the 30th International Conference on Automata, Languages and Programming (ICALP’03). Springer-Verlag, Berlin, 291--299. http://dl.acm.org/citation.cfm?id=1759210.1759241. Google Scholar
Digital Library
- Marvin Minsky and Seymour Papert. 1969. Perceptrons: An Introduction to Computational Geometry. MIT Press. Google Scholar
Digital Library
- Noam Nisan and Mario Szegedy. 1992. On the degree of Boolean functions as real polynomials. In Proceedings of the 24th Annual ACM Symposium on Theory of Computing (STOC’92). ACM, New York, NY, 462--467. DOI:https://doi.org/10.1145/129712.129757 Google Scholar
Digital Library
- Ramamohan Paturi. 1992. On the degree of polynomials that approximate symmetric Boolean functions. In Proceedings of the 24th Annual ACM Symposium on Theory of Computing (STOC’92). ACM, New York, NY, 468--474. DOI:https://doi.org/10.1145/129712.129758 Google Scholar
Digital Library
- Michael Saks and Avi Wigderson. 1986. Probabilistic Boolean decision trees and the complexity of evaluating game trees. In Proceedings of the 27th Annual Symposium on Foundations of Computer Science (SFCS’86). IEEE Computer Society, Washington, DC, 29--38. DOI:https://doi.org/10.1109/SFCS.1986.44 Google Scholar
Digital Library
- Alexander A. Sherstov. 2009. The intersection of two halfspaces has high threshold degree. In Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS’09). IEEE Computer Society, Washington, DC, 343--362. DOI:https://doi.org/10.1109/FOCS.2009.18 Google Scholar
Digital Library
- Alexander A. Sherstov. 2011. Strong direct product theorems for quantum communication and query complexity. In Proceedings of the 43rd Annual ACM Symposium on Theory of Computing (STOC’11). ACM, New York, NY, 41--50. DOI:https://doi.org/10.1145/1993636.1993643 Google Scholar
Digital Library
- Alexander A. Sherstov. 2012. Making polynomials robust to noise. In Proceedings of the 44th Annual ACM Symposium on Theory of Computing (STOC’12). ACM, New York, NY, 747--758. DOI:https://doi.org/10.1145/2213977.2214044 Google Scholar
Digital Library
- Alexander A. Sherstov. 2013. Approximating the AND-OR tree. Theory of Computing 9, 20 (2013), 653--663. DOI:https://doi.org/10.4086/toc.2013.v009a020Google Scholar
Cross Ref
- Alexander A. Sherstov and Pei Wu. 2019. Near-optimal lower bounds on the threshold degree and sign-rank of AC0. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (STOC’19). Association for Computing Machinery, New York, NY, 401--412. DOI:https://doi.org/10.1145/3313276.3316408 Google Scholar
Digital Library
- Yaoyun Shi. 2002. Approximating linear restrictions of Boolean functions. Retrieved from https://web.eecs.umich.edu/ shiyy/mypapers/linear02-j.ps.Google Scholar
Index Terms
Lower Bounding the AND-OR Tree via Symmetrization
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