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

Author Zachary Remscrim

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Zachary Remscrim
  • Department of Computer Science, The University of Chicago, IL, USA

Acknowledgements

The author would like to express his sincere gratitude to Professor Michael Sipser for many years of mentorship and support, without which this work would not have been possible, and to thank Professor Richard Lipton, as well as the anonymous reviewers, for several helpful comments on an earlier draft of this paper.

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Zachary Remscrim. Lower Bounds on the Running Time of Two-Way Quantum Finite Automata and Sublogarithmic-Space Quantum Turing Machines. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 39:1-39:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ITCS.2021.39

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
  • Theory of computation → Quantum computation theory
Keywords
  • Quantum computation
  • Lower bounds
  • Finite automata

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