The Structure of Promises in Quantum Speedups

Author Shalev Ben-David

Thumbnail PDF


  • Filesize: 482 kB
  • 14 pages

Document Identifiers

Author Details

Shalev Ben-David

Cite AsGet BibTex

Shalev Ben-David. The Structure of Promises in Quantum Speedups. In 11th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 61, pp. 7:1-7:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


In 1998, Beals, Buhrman, Cleve, Mosca, and de Wolf showed that no super-polynomial quantum speedup is possible in the query complexity setting unless there is a promise on the input. We examine several types of "unstructured" promises, and show that they also are not compatible with super-polynomial quantum speedups. We conclude that such speedups are only possible when the input is known to have some structure. Specifically, we show that there is a polynomial relationship of degree 18 between D(f) and Q(f) for any Boolean function f defined on permutations (elements of [n]^n in which each alphabet element occurs exactly once). More generally, this holds for all f defined on orbits of the symmetric group action (which acts on an element of [M]^n by permuting its entries). We also show that any Boolean function f defined on a "symmetric" subset of the Boolean hypercube has a polynomial relationship between R(f) and Q(f) - although in that setting, D(f) may be exponentially larger.
  • Quantum computing
  • quantum query complexity
  • decision tree complexity
  • lower bounds
  • quantum adversary method


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Scott Aaronson and Andris Ambainis. The need for structure in quantum speedups. arXiv preprint, 2009. Google Scholar
  2. Scott Aaronson and Andris Ambainis. Forrelation: A problem that optimally separates quantum from classical computing. In Proceedings of the 47th ACM Symposium on Theory of Computing (STOC 2015), pages 307-316, 2015. URL:
  3. Scott Aaronson and Shalev Ben-David. Sculpting quantum speedups. arXiv preprint, 2015. Google Scholar
  4. Andris Ambainis. Quantum lower bounds by quantum arguments. Journal of Computer and System Sciences, 64(4):750-767, 2002. URL:
  5. 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., URL:
  6. Joe P Buhler, Hendrik W Lenstra Jr, and Carl Pomerance. Factoring integers with the number field sieve. In The development of the number field sieve, pages 50-94. Springer, 1993. 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. URL:
  8. Richard Cleve. The query complexity of order-finding. Information and Computation, 192(2):162-171, 2004. Google Scholar
  9. Ashwin Nayak. Inverting a permutation is as hard as unordered search. arXiv preprint, 2010. Google Scholar
  10. 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:
  11. Daniel R. Simon. On the power of quantum computation. SIAM Journal on Computing, 26(5):1474-1483, 1997. URL: