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.
@InProceedings{bendavid:LIPIcs.TQC.2016.7, author = {Ben-David, Shalev}, title = {{The Structure of Promises in Quantum Speedups}}, booktitle = {11th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2016)}, pages = {7:1--7:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-019-4}, ISSN = {1868-8969}, year = {2016}, volume = {61}, editor = {Broadbent, Anne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2016.7}, URN = {urn:nbn:de:0030-drops-66882}, doi = {10.4230/LIPIcs.TQC.2016.7}, annote = {Keywords: Quantum computing, quantum query complexity, decision tree complexity, lower bounds, quantum adversary method} }
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