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DOI: 10.4230/LIPIcs.CCC.2022.28
URN: urn:nbn:de:0030-drops-165908
URL: https://drops.dagstuhl.de/opus/volltexte/2022/16590/
Bansal, Nikhil ; Sinha, Makrand ; de Wolf, Ronald
Influence in Completely Bounded Block-Multilinear Forms and Classical Simulation of Quantum Algorithms
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Abstract
The Aaronson-Ambainis conjecture (Theory of Computing '14) says that every low-degree bounded polynomial on the Boolean hypercube has an influential variable. This conjecture, if true, would imply that the acceptance probability of every d-query quantum algorithm can be well-approximated almost everywhere (i.e., on almost all inputs) by a poly(d)-query classical algorithm. We prove a special case of the conjecture: in every completely bounded degree-d block-multilinear form with constant variance, there always exists a variable with influence at least 1/poly(d). In a certain sense, such polynomials characterize the acceptance probability of quantum query algorithms, as shown by Arunachalam, Briët and Palazuelos (SICOMP '19). As a corollary we obtain efficient classical almost-everywhere simulation for a particular class of quantum algorithms that includes for instance k-fold Forrelation. Our main technical result relies on connections to free probability theory.
BibTeX - Entry
@InProceedings{bansal_et_al:LIPIcs.CCC.2022.28,
author = {Bansal, Nikhil and Sinha, Makrand and de Wolf, Ronald},
title = {{Influence in Completely Bounded Block-Multilinear Forms and Classical Simulation of Quantum Algorithms}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {28:1--28:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16590},
URN = {urn:nbn:de:0030-drops-165908},
doi = {10.4230/LIPIcs.CCC.2022.28},
annote = {Keywords: Aaronson-Ambainis conjecture, Quantum query complexity, Classical query complexity, Free probability, Completely bounded norm, Analysis of Boolean functions, Influence}
}
| Keywords: | Aaronson-Ambainis conjecture, Quantum query complexity, Classical query complexity, Free probability, Completely bounded norm, Analysis of Boolean functions, Influence | |
| Seminar: | 37th Computational Complexity Conference (CCC 2022) | |
| Issue date: | 2022 | |
| Date of publication: | 11.07.2022 |
