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: https://drops.dagstuhl.de/opus/volltexte/2021/13578/
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Remscrim, Zachary

Lower Bounds on the Running Time of Two-Way Quantum Finite Automata and Sublogarithmic-Space Quantum Turing Machines

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LIPIcs-ITCS-2021-39.pdf (0.6 MB)


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}
}

Keywords: Quantum computation, Lower bounds, Finite automata
Collection: 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)
Issue Date: 2021
Date of publication: 04.02.2021


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