arXiv:0809.0073v2 [quant-ph] 17 Nov 2008 Languages recognized with unbounded error by quantum finite automata Abuzer Yakaryılmaz and A.C. Cem Say Bo˜ gazi¸ ci University, Department of Computer Engineering, Bebek 34342 ˙ Istanbul, Turkey abuzer,say@boun.edu.tr ⋆ February 22, 2019 Abstract. We prove the following facts about the language recognition power of Kondacs- Watrous quantum finite automata in the unbounded error setting: One-way automata of this kind recognize all and only the stochastic languages. When the tape head is allowed two-way (or even “1.5-way”) movement, more languages become recognizable. This leads to the conclusion that quantum Turing machines are more powerful than probabilistic Turing machines when restricted to constant space bounds. 1 Introduction Several alternative models [1,2,4,10,13,18,20] of quantum finite automata (QFA’s) have been studied in the recent years. Most of the attention in this regard has been focused on the classes of languages recognized by these machines with bounded error [1, 3, 8, 10, 13, 15, 16]. In this paper, we examine the computational power of one of the most popular QFA models, the measure-many (Kondacs-Watrous) QFA, in the unbounded error setting. We give a complete characterization of the class of languages recognized by one-way QFA’s of this kind; they turn out to recognize all and only the stochastic languages. We also show that allowing the tape head to “stay put” for some steps during its left-to- right traversal of the input increases the language recognition power of these QFA’s. This contrasts the situation in the classical probabilistic models, where two-way and one-way automata are equivalent in power in this setting [12]. We conclude that quantum Turing machines are strictly more powerful than their probabilistic counterparts when restricted to constant space bounds. The rest of this paper is structured as follows: Section 2 contains the relevant background informa- tion. Section 3 presents our results. Section 4 is a conclusion. 2 Preliminaries In this section, we give a brief review of the definitions and facts that will be used in the rest of the paper. 2.1 Classical Automata A 1-way probabilistic finite automaton (1pfa) [23] with n ∈ Z + states is a 5 tuple P =(S,Σ, {A σ | σ ∈ Σ}, v 0 ,F ), where 1. S = {s 1 , ··· ,s n } is the set of states, 2. Σ is the finite input alphabet, 3. A σ is the n × n real-valued stochastic transition matrix for symbol σ, that is, A σ (i,j ) is the value of the transition probability from state s i to state s j when reading symbol σ, ⋆ This work was partially supported by the Bo˜ gazi¸ ci University Research Fund with grant 08A102.