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.2021.39
URN: urn:nbn:de:0030-drops-135781
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/13578/
Remscrim, Zachary
Lower Bounds on the Running Time of Two-Way Quantum Finite Automata and Sublogarithmic-Space Quantum Turing Machines
Abstract
The two-way finite automaton with quantum and classical states (2QCFA), defined by Ambainis and Watrous, is a model of quantum computation whose quantum part is extremely limited; however, as they showed, 2QCFA are surprisingly powerful: a 2QCFA with only a single-qubit can recognize the language L_{pal} = {w ∈ {a,b}^*:w is a palindrome} with bounded error in expected time 2^{O(n)}.
We prove that their result cannot be improved upon: a 2QCFA (of any size) cannot recognize L_{pal} with bounded error in expected time 2^{o(n)}. This is the first example of a language that can be recognized with bounded error by a 2QCFA in exponential time but not in subexponential time. Moreover, we prove that a quantum Turing machine (QTM) running in space o(log n) and expected time 2^{n^{1-Ω(1)}} cannot recognize L_{pal} with bounded error; again, this is the first lower bound of its kind. Far more generally, we establish a lower bound on the running time of any 2QCFA or o(log n)-space QTM that recognizes any language L in terms of a natural "hardness measure" of L. This allows us to exhibit a large family of languages for which we have asymptotically matching lower and upper bounds on the running time of any such 2QCFA or QTM recognizer.
BibTeX - Entry
@InProceedings{remscrim:LIPIcs.ITCS.2021.39,
author = {Zachary Remscrim},
title = {{Lower Bounds on the Running Time of Two-Way Quantum Finite Automata and Sublogarithmic-Space Quantum Turing Machines}},
booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
pages = {39:1--39:20},
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/2021/13578},
URN = {urn:nbn:de:0030-drops-135781},
doi = {10.4230/LIPIcs.ITCS.2021.39},
annote = {Keywords: Quantum computation, Lower bounds, Finite automata}
}
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Keywords: |
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Quantum computation, Lower bounds, Finite automata |
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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 |
