Characterizations of 1-Way Quantum Finite Automata

Alex Brodsky, Nicholas Pippenger
2002 SIAM journal on computing (Print)  
The 2-way quantum finite automaton introduced by Kondacs and Watrous [KW97] can accept non-regular languages with bounded error in polynomial time. If we restrict the head of the automaton to moving classically and to moving only in one direction, the acceptance power of this 1-way quantum finite automaton is reduced to a proper subset of the regular languages. In this paper we study two different models of 1-way quantum finite automata. The first model, termed measure-once quantum finite
more » ... ta, was introduced by Moore and Crutchfield [MC00] , and the second model, termed measure-many quantum finite automata, was introduced by Kondacs and Watrous [KW97] . We characterize the measure-once model when it is restricted to accepting with bounded error and show that, without that restriction, it can solve the word problem over the free group. We also show that it can be simulated by a probabilistic finite automaton and describe an algorithm that determines if two measure-once automata are equivalent. We prove several closure properties of the classes of languages accepted by measuremany automata, including inverse homomorphisms, and provide a new necessary condition for a language to be accepted by the measure-many model with bounded error. Finally, we show that piecewise testable languages can be accepted with bounded error by a measure-many quantum finite automaton, in the process introducing new construction techniques for quantum automata. right on each transition, we get the 1-way quantum finite automaton of Kondacs and Watrous [KW97] , which, when accepting with bounded error, can only accept a proper subset of the regular languages. If the reading head is classical then quantum mechanical evolution hinders language acceptance; restricting the set of languages accepted by 1-way quantum finite automata with bounded error to a proper subset of the regular languages [KW97] . During its computation, a 1-way QFA performs measurements on its configuration. Since the acceptance capability of a 1-way QFA depends on the measurements that the QFA may perform during the computation, we investigate two models of 1-way QFAs that differ only in the type of measurement that they perform during the computation. The first model, termed measure-once quantum finite automata (MO-QFAs), is similar to the one introduced by Moore and Crutchfield[MC00]. The second model, termed measure-many quantum finite automata (MM-QFAs), is similar to the one introduced by Kondacs and Watrous[KW97], and is more complex than the MO-QFA. The main difference between the two models is that a measure-once automaton performs one measurement at the end of its computation, while a measure-many automaton performs a measurement after every transition. This makes the measure-many model more powerful than the measure-once model, where the power of a model refers to the acceptance capability of the corresponding automata. First, we present results dealing with MO-QFAs. We show that the class of languages accepted by MO-QFAs with bounded error is exactly the class of group languages. Consequently, this class of languages accepted by MO-QFAs is closed under inverse homomorphisms, word quotients, and boolean operations. We show that MO-QFAs that do not accept with bounded error can accept non-regular languages and, in particular, can solve the word problem over the free group. We also describe an algorithm that determines if two MO-QFAs are equivalent and prove that probabilistic finite automata (PFAs) can simulate MO-QFAs. Second, we shift our focus to MM-QFAs. We show that the classes of languages accepted by these automata are closed under complement, inverse homomorphisms, and word quotients. We prove by example that the class of languages accepted by MM-QFAs with bounded error is not closed under homomorphisms, and prove a necessary condition for membership within this class. We also relate the sufficiency of this condition to the question of whether the class is closed under boolean operations. Finally, we show, by construction, that MM-QFAs can accept piecewise testable languages with bounded error and introduce novel concepts for constructing MM-QFAs. The rest of the paper is organized in the following way: Section 2 contains the definitions of the quantum automata and background information, Section 3 discusses measure-once quantum finite automata, Section 4 discusses measure-many quantum finite automata, and Section 5 summarizes. Definitions and Background Definition of MO-QFA A measure-once quantum finite automaton is defined by a 5-tuple M = (Q, Σ, δ, q 0 , F )
doi:10.1137/s0097539799353443 fatcat:uizartdqxzblbbg64uxb5hnx2y