It is known since the work of [Aaronson and Ambainis, 2014] that for any permutation symmetric function f, the quantum query complexity is at most polynomially smaller than the classical randomized query complexity, more precisely that R(f) = O~(Q^7(f)). In this paper, we improve this result and show that R(f) = O(Q^3(f)) for a more general class of symmetric functions. Our proof is constructive and relies largely on the quantum hardness of distinguishing a random permutation from a random function with small range from Zhandry [Zhandry, 2015].
@InProceedings{chailloux:LIPIcs.ITCS.2019.19, author = {Chailloux, Andr\'{e}}, title = {{A Note on the Quantum Query Complexity of Permutation Symmetric Functions}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {19:1--19:7}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-095-8}, ISSN = {1868-8969}, year = {2019}, volume = {124}, editor = {Blum, Avrim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.19}, URN = {urn:nbn:de:0030-drops-101126}, doi = {10.4230/LIPIcs.ITCS.2019.19}, annote = {Keywords: quantum query complexity, permutation symmetric functions} }
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