Adaptivity vs. Postselection, and Hardness Amplification for Polynomial Approximation

Author Lijie Chen

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Lijie Chen

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Lijie Chen. Adaptivity vs. Postselection, and Hardness Amplification for Polynomial Approximation. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 26:1-26:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


We study the following problem: with the power of postselection (classically or quantumly), what is your ability to answer adaptive queries to certain languages? More specifically, for what kind of computational classes C, we have P^C belongs to PostBPP or PostBQP? While a complete answer to the above question seems impossible given the development of present computational complexity theory. We study the analogous question in query complexity, which sheds light on the limitation of relativized methods (the relativization barrier) to the above question. Informally, we show that, for a partial function f, if there is no efficient small bounded-error algorithm for f classically or quantumly, then there is no efficient postselection bounded-error algorithm to answer adaptive queries to f classically or quantumly. Our results imply a new proof for the classical oracle separation P^{NP^O} notsubset PP^O, which is arguably more elegant. They also lead to a new oracle separation P^{SZK^O} notsubset PP^O, which is close to an oracle separation between SZK and PP - an open problem in the field of oracle separations. Our result also implies a hardness amplification construction for polynomial approximation: given a function f on n bits, we construct an adaptive-version of f, denoted by F, on O(m·n) bits, such that if f requires large degree to approximate to error 2/3 in a certain one-sided sense, then F requires large degree to approximate even to error 1/2 - 2^{-m}. Our construction achieves the same amplification in the work of Thaler (ICALP, 2016), by composing a function with O(log n) deterministic query complexity, which is in sharp contrast to all the previous results where the composing amplifiers are all hard functions in a certain sense.
  • approximate degree
  • postselection
  • hardness amplification
  • adaptivity


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  1. Scott Aaronson. Quantum computing, postselection, and probabilistic polynomial-time. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 461(2063):3473-3482, 2005. Google Scholar
  2. Scott Aaronson. Impossibility of succinct quantum proofs for collision-freeness. Quantum Information &Computation, 12(1-2):21-28, 2012. Google Scholar
  3. 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. ACM, 2011. Google Scholar
  4. Richard Beigel. Perceptrons, PP, and the polynomial hierarchy. Computational Complexity, 4(4):339-349, 1994. Google Scholar
  5. Adam Bouland, Lijie Chen, Dhiraj Holden, Justin Thaler, and Prashant Nalini Vasudevan. On SZK and PP. In Electronic Colloquium on Computational Complexity (ECCC), volume 23, page 140, 2016. Google Scholar
  6. Michael J. Bremner, Richard Jozsa, and Dan J. Shepherd. Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 467(2126):459-472, 2011. Google Scholar
  7. Harry Buhrman and Ronald De Wolf. Complexity measures and decision tree complexity: a survey. Theoretical Computer Science, 288(1):21-43, 2002. Google Scholar
  8. Mark Bun and Justin Thaler. Dual polynomials for collision and element distinctness. arXiv preprint arXiv:1503.07261, 2015. Google Scholar
  9. Mark Bun and Justin Thaler. Hardness amplification and the approximate degree of constant-depth circuits. In International Colloquium on Automata, Languages, and Programming, pages 268-280. Springer, 2015. Google Scholar
  10. Mark Bun and Justin Thaler. Approximate degree and the complexity of depth three circuits. In Electronic Colloquium on Computational Complexity (ECCC), volume 23, page 121, 2016. Google Scholar
  11. Andrew Drucker and Ronald de Wolf. Quantum proofs for classical theorems. arXiv preprint arXiv:0910.3376, 2009. Google Scholar
  12. Yenjo Han, Lane A Hemaspaandra, and Thomas Thierauf. Threshold computation and cryptographic security. SIAM Journal on Computing, 26(1):59-78, 1997. Google Scholar
  13. Greg Kuperberg. How hard is it to approximate the Jones polynomial? arXiv preprint arXiv:0908.0512, 2009. Google Scholar
  14. Noam Nisan and Mario Szegedy. On the degree of boolean functions as real polynomials. Computational complexity, 4(4):301-313, 1994. Google Scholar
  15. Alexander A. Sherstov. Breaking the Minsky-Papert barrier for constant-depth circuits. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, pages 223-232. ACM, 2014. Google Scholar
  16. Alexander A Sherstov. The power of asymmetry in constant-depth circuits. In Foundations of Computer Science (FOCS), 2015 IEEE 56th Annual Symposium on, pages 431-450. IEEE, 2015. Google Scholar
  17. Justin Thaler. Lower bounds for the approximate degree of block-composed functions. In Electronic Colloquium on Computational Complexity (ECCC), volume 21, page 150, 2014. Google Scholar
  18. NK Vereschchagin. On the power of pp. In Structure in Complexity Theory Conference, 1992., Proceedings of the Seventh Annual, pages 138-143. IEEE, 1992. Google Scholar
  19. Mikhail N. Vyalyi. QMA = PP implies that PP contains PH. In ECCCTR: Electronic Colloquium on Computational Complexity, technical reports, 2003. URL:
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