Quantum Finite Automata and Weighted Automata

Quantum finite automata derive their strength by exploiting interference in complex valued probability amplitudes. Of particular interest is the 2-way model of Ambainis and Watrous that has both quantum and classical states (2QCFA) [A. Ambainis and J. Watrous, Two-way finite automata with quantum and classical state, Theoretical Computer Science, 287(1), pp. 299-311, 2002], since it combines the advantage of the power of interference in a constant-sized quantum system with a 2-way head. This paper is a step towards finding the least powerful model which is purely classical and can mimic the dynamics of quantum phase. We consider weighted automata with the Cortes-Mohri definition of language recognition [C. Cortes and M. Mohri, Context-Free Recognition with Weighted Automata, Grammars 3(2/3), pp. 133-150, 2000] as a candidate model for simulating 2QCFA. Given any 2QCFA that (i) uses the accept-reject-continue observable, (ii) recognizes a language with one-sided error and (iii) the entries of whose unitary matrices are algebraic complex numbers, we show a method of constructing a weighted automaton over $\mathbb{C}$ that simulates it efficiently.Comment: 10 pages, Preliminary version appears in the Proceedings of ACiD-2005, Texts in Algorithmics series of KCL publications, pp. 123-134, 200

Exact Quantum Algorithms for the Leader Election Problem

This paper gives the first separation of quantum and classical pure (i.e., non-cryptographic) computing abilities with no restriction on the amount of available computing resources, by considering the exact solvability of a celebrated unsolvable problem in classical distributed computing, the ``leader election problem'' on anonymous networks. The goal of the leader election problem is to elect a unique leader from among distributed parties. The paper considers this problem for anonymous networks, in which each party has the same identifier. It is well-known that no classical algorithm can solve exactly (i.e., in bounded time without error) the leader election problem in anonymous networks, even if it is given the number of parties. This paper gives two quantum algorithms that, given the number of parties, can exactly solve the problem for any network topology in polynomial rounds and polynomial communication/time complexity with respect to the number of parties, when the parties are connected by quantum communication links.Comment: 47 pages, preliminary version in Proceedings of STACS 200

Quantum Finite Automata and Weighted Automata

Abstract. Quantum finite automata derive their strength by exploiting interference in complex valued probability amplitudes. Of particular interest is the 2-way model of Ambainis and Watrous that has both quantum and classical states (2QCFA) [A. Ambainis and J. Watrous, Two-way finite automata with quantum and classical state, Theoretical Computer Science, 287(1), pp. 299-311, 2002], since it combines the advantage of the power of interference in a constant-sized quantum system with a 2-way head. This paper is a step towards finding the least powerful model which is purely classical and can mimic the dynamics of quantum phase. We consider weighted automata with the Cortes-Mohri definition of language recognition [C. Cortes and M. Mohri, Context-Free Recognition with Weighted Automata, Grammars 3(2/3), pp. 133-150, 2000] as a candidate model for simulating 2QCFA. Given any 2QCFA that (i) uses the accept-reject-continue observable, (ii) recognizes a language with one-sided error and (iii) the entries of whose unitary matrices are algebraic complex numbers, we show a method of constructing a weighted automaton over C that simulates it efficiently.