Efficient probability amplification in two-way quantum finite automata

In classical computation, one only needs to sequence 0(log E) identical copies of a given probabilistic automaton with one-sided error p < 1 to run on the same input in order to obtain a two-way machine with error bound epsilon. For two-way quantum finite automata (2qfa's), this straightforward approach does not yield efficient results; the number of machine copies required to reduce the error to epsilon can be as high as (1/epsilon)(2). In their celebrated proof that 2qfa's can recognize the non-regular language L = {a(n)b(n) vertical bar n > 0}, Kondacs and Watrous use a different probability amplification method, which yields machines with O((1/epsilon)(2)) states, and with runtime O(1/epsilon vertical bar omega vertical bar), where omega is the input string. In this paper, we examine significantly more efficient techniques of probability amplification. One of our methods produces machines which decide L in O(vertical bar omega vertical bar) time (i.e. the running time does not depend on the error bound) and which have O((1/epsilon)(2/c)) states for any given constant c > 1. Other methods, yielding machines whose state complexities are polylogarithmic in 1/epsilon, including one which halts in o(log(1/epsilon)vertical bar omega vertical bar) time, are also presented. (C) 2009 Elsevier B.V. All rights reserved.