Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH scholarly article en Mudigonda, Abhijit S.; Williams, R. Ryan https://www.dagstuhl.de/lipics License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
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URN: urn:nbn:de:0030-drops-135897
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Time-Space Lower Bounds for Simulating Proof Systems with Quantum and Randomized Verifiers

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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}
}

Keywords: Time-space tradeoffs, lower bounds, QMA
Seminar: 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)
Issue date: 2021
Date of publication: 04.02.2021


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