Small size quantum automata recognizing some regular languages

  • Authors:

  • Alberto Bertoni;Carlo Mereghetti;Beatrice Palano

  • Affiliations:

  • Dipartimento di Scienze dell'Informazione, Università degli Studi di Milano, Milano, Italy;Dipartimento di Scienze dell'Informazione, Università degli Studi di Milano, Milano, Italy;Dipartimento di Scienze dell'Informazione, Università degli Studi di Milano, Milano, Italy

  • Venue:
  • Theoretical Computer Science - The art of theory
  • Year:
  • 2005

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Abstract

Given a class {pα | α ∈ I} of stochastic events induced by M-state 1-way quantum finite automata (1qfa) on alphabet Σ, we investigate the size (number of states) of 1qfa's that δ-approximate a convex linear combination of {pα | α ∈ I}, and we apply the results to the synthesis of small size 1qfa's. We obtain: • An O((Md/δ3) log2(d/δ2)) general size bound, where d is the Vapnik dimension of {pα(w) | w ∈ Σ*}. • For commutative n-periodic events p on Σ with |Σ| = H, we prove an O((H log n/δ2)) size bound for inducing a δ-approximation of ½ + ½ p whenever ||F(p)||1 ≤nH, where F(p) is the discrete Fourier transform of (the vector p associated with) p. • If the characteristic function χL of an n-periodic unary language L satisfies ||F(χL))||1 ≤ n, then L is recognized with isolated cut-point by a 1qfa with O(log n) states. Vice versa, if L is recognized with isolated cut-point by a 1qfa with O(log n) state, then ||F(χL))||1 = O(n log n).