5 research outputs found
Potential of quantum finite automata with exact acceptance
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 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 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.Comment: We have improved the presentation of the paper. Accepted to
International Journal of Foundation of Computer Scienc
Photonic realization of a quantum finite automaton
We describe a physical implementation of a quantum finite automaton that recognizes a well-known family of periodic languages. The realization exploits the polarization degree of freedom of single photons and their manipulation through linear optical elements. We use techniques of confidence amplification to reduce the acceptance error probability of the automaton. It is worth remarking that the quantum finite automaton we physically realize is not only interesting per se but it turns out to be a crucial building block in many quantum finite automaton design frameworks theoretically settled in the literature
Golomb rulers and difference sets for succinct quantum automata
Given a function p : N \u2192 [0,1] of period n, we study the minimal size (number of states) of a one-way quantum finite automaton (Iqfa) inducing the stochastic event ap + b, for real constants a>0, b 650, a+b 641. First of all, we relate the estimation of the minimal size to the problem of finding a minimal difference cover for a suitable subset of Zn. Then, by observing that the cardinality of a difference cover \u394 for a set A Z n, must satisfy , we investigate the class of sets A admitting difference covers of cardinality exactly . We relate this problem with the efficient construction of Golomb rulers and difference sets. We design an algorithm which outputs each of the Golomb rulers (if any) of a given set in pseudo-polynomial time. As a consequence, we obtain an efficient algorithm that construct minimal difference covers for a non-trivial class of sets. Moreover, by using projective geometry arguments, we give an algorithm that, for any n=q2+q+1 with q prime power, constructs difference sets for Z n in quadratic time