License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CCC.2016.26
URN: urn:nbn:de:0030-drops-58538
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/5853/
Aaronson, Scott ;
Ben-David, Shalev
Sculpting Quantum Speedups
Abstract
Given a problem which is intractable for both quantum and classical algorithms, can we find a sub-problem for which quantum algorithms provide an exponential advantage? We refer to this problem as the "sculpting problem." In this work, we give a full characterization of sculptable functions in the query complexity setting. We show that a total function f can be restricted to a promise P such that Q(f|_P)=O(polylog(N)) and R(f|_P)=N^{Omega(1)}, if and only if f has a large number of inputs with large certificate complexity. The proof uses some interesting techniques: for one direction, we introduce new relationships between randomized and quantum query complexity in various settings, and for the other direction, we use a recent result from communication complexity due to Klartag and Regev. We also characterize sculpting for other query complexity measures, such as R(f) vs. R_0(f) and R_0(f) vs. D(f).
Along the way, we prove some new relationships for quantum query complexity: for example, a nearly quadratic relationship between Q(f) and D(f) whenever the promise of f is small. This contrasts with the recent super-quadratic query complexity separations, showing that the maximum gap between classical and quantum query complexities is indeed quadratic in various settings - just not for total functions!
Lastly, we investigate sculpting in the Turing machine model. We show that if there is any BPP-bi-immune language in BQP, then every language outside BPP can be restricted to a promise which places it in PromiseBQP but not in PromiseBPP. Under a weaker assumption, that some problem in BQP is hard on average for P/poly, we show that every paddable language outside BPP is sculptable in this way.
BibTeX - Entry
@InProceedings{aaronson_et_al:LIPIcs:2016:5853,
author = {Scott Aaronson and Shalev Ben-David},
title = {{Sculpting Quantum Speedups}},
booktitle = {31st Conference on Computational Complexity (CCC 2016)},
pages = {26:1--26:28},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-008-8},
ISSN = {1868-8969},
year = {2016},
volume = {50},
editor = {Ran Raz},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/5853},
URN = {urn:nbn:de:0030-drops-58538},
doi = {10.4230/LIPIcs.CCC.2016.26},
annote = {Keywords: Quantum Computing, Query Complexity, Decision Tree Complexity, Structural Complexity}
}
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Keywords: |
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Quantum Computing, Query Complexity, Decision Tree Complexity, Structural Complexity |
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Collection: |
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31st Conference on Computational Complexity (CCC 2016) |
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Issue Date: |
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2016 |
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Date of publication: |
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19.05.2016 |
