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.ITCS.2021.50
URN: urn:nbn:de:0030-drops-135897
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/13589/
Mudigonda, Abhijit S. ;
Williams, R. Ryan
Time-Space Lower Bounds for Simulating Proof Systems with Quantum and Randomized Verifiers
Abstract
A line of work initiated by Fortnow in 1997 has proven model-independent time-space lower bounds for the SAT problem and related problems within the polynomial-time hierarchy. For example, for the SAT problem, the state-of-the-art is that the problem cannot be solved by random-access machines in n^c time and n^o(1) space simultaneously for c < 2cos(π/7) ≈ 1.801.
We extend this lower bound approach to the quantum and randomized domains. Combining Grover’s algorithm with components from SAT time-space lower bounds, we show that there are problems verifiable in O(n) time with quantum Merlin-Arthur protocols that cannot be solved in n^c time and n^o(1) space simultaneously for c < (3+√3)/2 ≈ 2.366, a super-quadratic time lower bound. This result and the prior work on SAT can both be viewed as consequences of a more general formula for time lower bounds against small-space algorithms, whose asymptotics we study in full.
We also show lower bounds against randomized algorithms: there are problems verifiable in O(n) time with (classical) Merlin-Arthur protocols that cannot be solved in n^c randomized time and O(log n) space simultaneously for c < 1.465, improving a result of Diehl. For quantum Merlin-Arthur protocols, the lower bound in this setting can be improved to c < 1.5.
BibTeX - Entry
@InProceedings{mudigonda_et_al:LIPIcs.ITCS.2021.50,
author = {Abhijit S. Mudigonda and R. Ryan Williams},
title = {{Time-Space Lower Bounds for Simulating Proof Systems with Quantum and Randomized Verifiers}},
booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
pages = {50:1--50:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-177-1},
ISSN = {1868-8969},
year = {2021},
volume = {185},
editor = {James R. Lee},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/13589},
URN = {urn:nbn:de:0030-drops-135897},
doi = {10.4230/LIPIcs.ITCS.2021.50},
annote = {Keywords: Time-space tradeoffs, lower bounds, QMA}
}
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Keywords: |
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Time-space tradeoffs, lower bounds, QMA |
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Collection: |
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12th Innovations in Theoretical Computer Science Conference (ITCS 2021) |
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Issue Date: |
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2021 |
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Date of publication: |
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04.02.2021 |
