Small size quantum automata recognizing some regular languages

Abstract

Given a class { p α | 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 { p α | α ∈ I } , and we apply the results to the synthesis of small size 1qfa's. We obtain: • An O ( ( Md / δ 3 ) log 2 ( d / δ 2 ) ) general size bound, where d is the Vapnik dimension of { p α ( w ) | w ∈ Σ * } . • For commutative n-periodic events p on Σ with | Σ | = H , we prove an O ( ( H log n / δ 2 ) ) size bound for inducing a δ -approximation of 1 2 + 1 2 p whenever ∥ F ( p ^ ) ∥ 1 ⩽ n H , where F ( p ^ ) is the discrete Fourier transform of (the vector p ^ associated with) p. • If the characteristic function χ L of an n-periodic unary language L satisfies ∥ F ( χ L ^ ) ) ∥ 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 ( χ L ^ ) ) ∥ 1 = O ( n log n ) .