Potential of Quantum Finite Automata with Exact Acceptance

Abstract

The potential of the exact quantum information processing is an interesting, important and intriguing issue. For examples, it has been believed that quantum tools can provide significant, that is larger than polynomial, advantages in the case of exact quantum computation only, or mainly, for problems with very special structures. We will show that this is not the case.In this paper the potential of quantum finite automata producing outcomes not only with a (high) probability, but with certainty (so called exactly) is explored in the context of their uses for solving promise problems and with respect to the size of automata. It is shown that for solving particular classes {A n } ∞ n=1 of promise problems, even those without some very special structure, that succinctness of the exact quantum finite automata under consideration, with respect to the number of (basis) states, can be very small (and constant) though it grows proportional to n in the case deterministic finite automata (DFAs) of the same power are used. This is here demonstrated also for the case that the component languages of the promise problems solvable by DFAs are non-regular. The method used can be applied in finding more exact quantum finite automata or quantum algorithms for other promise problems.

Promise problems solved by quantum and classical finite automata

“…can solve the promise problem A N, l . Therefore, the minimal DFA to solve the promise problem A N, l has less than N states, contradicting the fact that the minimal DFA to solve A N, l has N states [18].…”

“…where l is a fix positive integer. The promise problem B l can be solved by a DFA [18]. Therefore it can be solved by a pvDFA.…”

“…It is easy to see that s(A N, l yes ) = N and s(A N, l no ) = N . It has been proved in [18] that ss(A N, l ) = N . Therefore ss(A N, l ) = min{s(A N, l yes ), s(A N, l no )} = N .…”

Quantum Finite Automata: A Modern Introduction

“…We also refer the reader to the recent works by Gruska and colleagues [14,15,28] for further results on the succinctness of exact QFAs.…”

Unary Probabilistic and Quantum Automata on Promise Problems

“…An initial result was given to compare exact quantum and deterministic pushdown automata [19], the former one was shown to be more powerful (see also [20] and [21] for the results in this direction). Then, the result given by Ambainis and Yakaryılmaz [6], the state advantages of exact quantum finite automata (QFAs) over deterministic finite automata (DFAs) cannot be bounded in the case of unary promise problems, has stimulated the topic and a series of papers appeared on the succinctness of QFAs and other models [30,12,13,29,8,1]. In parallel, the new results were given on classical and quantum automata models [23,10]:…”