首页    期刊浏览 2024年11月29日 星期五
登录注册

文章基本信息

  • 标题:The Limitations of Few Qubits: One-way and Two-way Quantum Finite Automata and the Group Word Problem
  • 本地全文:下载
  • 作者:Zachary Remscrim
  • 期刊名称:Electronic Colloquium on Computational Complexity
  • 印刷版ISSN:1433-8092
  • 出版年度:2019
  • 卷号:2019
  • 页码:1-48
  • 出版社:Universität Trier, Lehrstuhl für Theoretische Computer-Forschung
  • 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 pa l = w a b : w is a palindrome with bounded-error in expected exponential time. We prove that their result essentially cannot be improved upon: a 2QCFA (of any finite size) cannot recognize L pa l with bounded-error in expected time 2 o ( n ) , on inputs of length n . To our knowledge, 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. A key tool in our result is a generalization to 2QCFA of a technical lemma that was used by Dwork and Stockmeyer to prove a lower bound on the expected running time of any two-way probabilistic finite automaton that recognizes a non-regular language with bounded-error.

    Furthermore, we prove strong lower bounds on the expected running time of any 2QCFA that recognizes a group word problem with bounded-error. In a recent paper, we showed that 2QCFA can recognize, with bounded-error, a broad class of group word problems in expected exponential time, and a more narrow class of group word problems in expected polynomial time. As a consequence, we can now exhibit a large family of natural languages that can be recognized with bounded-error by a 2QCFA in expected exponential time, but not in expected subexponential time. Moreover, we obtain significant progress towards a precise classification of those group word problems that can be recognized with bounded-error in expected polynomial time by a 2QCFA.We also consider the one-way measure-once quantum finite automaton (1QFA), defined by Moore and Crutchfield, as well as a natural generalization to one-way measure-once finite automata with quantum and classical states (1QCFA). We precisely classify those groups whose word problem may be recognized with positive one-sided error (for both the bounded-error and unbounded-error cases) by a 1QFA or 1QCFA with any particular number of quantum states and any particular number of classical states; we also obtain partial results in the negative one-sided error case. As an immediate corollary, we show that allowing a 1QFA or 1QCFA to have even a single additional quantum or classical state enlarges the class of languages that may be recognized with positive one-sided error (of either type).

  • 关键词:finite automata ; lower bound ; quantum ; word problem
国家哲学社会科学文献中心版权所有