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Quantum State Complexity of Formal Languages

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Descriptional Complexity of Formal Systems (DCFS 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9118))

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Abstract

In this extended abstract, our notion of state complexity concerns the minimal amount of descriptive information necessary for a finite automaton to determine whether given fixed-length strings belong to a target language. This serves as a descriptional complexity measure for languages with respect to input length. In particular, we study the minimal number of inner states of quantum finite automata, whose tape heads may move freely in all directions and which conduct a projective measurement at every step, to recognize given languages. Such a complexity measure is referred to as the quantum state complexity of languages. We demonstrate upper and lower bounds on the quantum state complexity of languages on various types of quantum finite automata. By inventing a notion of timed crossing sequence, we also establish a general lower-bound on quantum state complexity in terms of approximate matrix rank. As a consequence, we show that bounded-error 2qfa’s running in expected subexponential time cannot, in general, simulate logarithmic-space deterministic Turing machines.

M. Villagra is a research fellow of the Japan Society for the Promotion of Sciences (JSPS).

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Notes

  1. 1.

    Traditionally, “state complexity” refers to the minimal descriptional size of finite automata that recognize a language on all inputs and this notion has been proven to be useful to study the complexity of regular languages.

  2. 2.

    This model is sometimes referred to as “real time” because its tape head always moves to the right without staying still on any tape cell.

  3. 3.

    In other words, \(M\) recognizes a so-called “promise problem” \((L_n,\varSigma ^n-L_n)\) with error probability at most \(\varepsilon (n)\).

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Authors and Affiliations

  1. Department of Information Science, University of Fukui, 3-9-1 Bunkyo, Fukui, 910-8507, Japan

    Marcos Villagra & Tomoyuki Yamakami

Authors
  1. Marcos Villagra
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  2. Tomoyuki Yamakami
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Correspondence to Tomoyuki Yamakami .

Editor information

Editors and Affiliations

  1. School of Computer Science, University of Waterloo, Waterloo, Ontario, Canada

    Jeffrey Shallit

  2. Department of Mathematics, University of Turku, Turku, Finland

    Alexander Okhotin

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Villagra, M., Yamakami, T. (2015). Quantum State Complexity of Formal Languages. In: Shallit, J., Okhotin, A. (eds) Descriptional Complexity of Formal Systems. DCFS 2015. Lecture Notes in Computer Science(), vol 9118. Springer, Cham. https://doi.org/10.1007/978-3-319-19225-3_24

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  • DOI: https://doi.org/10.1007/978-3-319-19225-3_24

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  • Publisher Name: Springer, Cham

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  • Online ISBN: 978-3-319-19225-3

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