An Application of Quantum Finite Automata to Interactive Proof Systems

Quantum finite automata have been studied intensively since their introduction in late 1990s as a natural model of a quantum computer with finite-dimensional quantum memory space. This paper seeks their direct application to interactive proof systems in which a mighty quantum prover communicates with a quantum-automaton verifier through a common communication cell. Our quantum interactive proof systems are juxtaposed to Dwork-Stockmeyer's classical interactive proof systems whose verifiers are two-way probabilistic automata. We demonstrate strengths and weaknesses of our systems and further study how various restrictions on the behaviors of quantum-automaton verifiers affect the power of quantum interactive proof systems.Comment: This is an extended version of the conference paper in the Proceedings of the 9th International Conference on Implementation and Application of Automata, Lecture Notes in Computer Science, Springer-Verlag, Kingston, Canada, July 22-24, 200

Finite state verifiers with constant randomness

We give a new characterization of $\mathsf{NL}$ as the class of languages whose members have certificates that can be verified with small error in polynomial time by finite state machines that use a constant number of random bits, as opposed to its conventional description in terms of deterministic logarithmic-space verifiers. It turns out that allowing two-way interaction with the prover does not change the class of verifiable languages, and that no polynomially bounded amount of randomness is useful for constant-memory computers when used as language recognizers, or public-coin verifiers. A corollary of our main result is that the class of outcome problems corresponding to O(log n)-space bounded games of incomplete information where the universal player is allowed a constant number of moves equals NL.Comment: 17 pages. An improved versio

One-Way Reversible and Quantum Finite Automata with Advice

We examine the characteristic features of reversible and quantum computations in the presence of supplementary external information, known as advice. In particular, we present a simple, algebraic characterization of languages recognized by one-way reversible finite automata augmented with deterministic advice. With a further elaborate argument, we prove a similar but slightly weaker result for bounded-error one-way quantum finite automata with advice. Immediate applications of those properties lead to containments and separations among various language families when they are assisted by appropriately chosen advice. We further demonstrate the power and limitation of randomized advice and quantum advice when they are given to one-way quantum finite automata.Comment: A4, 10pt, 1 figure, 31 pages. This is a complete version of an extended abstract appeared in the Proceedings of the 6th International Conference on Language and Automata Theory and Applications (LATA 2012), March 5-9, 2012, A Coruna, Spain, Lecture Notes in Computer Science, Springer-Verlag, Vol.7183, pp.526-537, 201

On the state complexity of semi-quantum finite automata

Some of the most interesting and important results concerning quantum finite automata are those showing that they can recognize certain languages with (much) less resources than corresponding classical finite automata \cite{Amb98,Amb09,AmYa11,Ber05,Fre09,Mer00,Mer01,Mer02,Yak10,ZhgQiu112,Zhg12}. This paper shows three results of such a type that are stronger in some sense than other ones because (a) they deal with models of quantum automata with very little quantumness (so-called semi-quantum one- and two-way automata with one qubit memory only); (b) differences, even comparing with probabilistic classical automata, are bigger than expected; (c) a trade-off between the number of classical and quantum basis states needed is demonstrated in one case and (d) languages (or the promise problem) used to show main results are very simple and often explored ones in automata theory or in communication complexity, with seemingly little structure that could be utilized.Comment: 19 pages. We improve (make stronger) the results in section