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  • 标题:Lower Bounds on the Running Time of Two-Way Quantum Finite Automata and Sublogarithmic-Space Quantum Turing Machines
  • 本地全文:下载
  • 作者:Zachary Remscrim
  • 期刊名称:LIPIcs : Leibniz International Proceedings in Informatics
  • 电子版ISSN:1868-8969
  • 出版年度:2021
  • 卷号:185
  • 页码:39:1-39:20
  • DOI:10.4230/LIPIcs.ITCS.2021.39
  • 出版社:Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
  • 摘要: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 â^^ {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.
  • 关键词:Quantum computation; Lower bounds; Finite automata
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