Efficient probability amplification in two-way quantum finite automata (original) (raw)

In classical computation, one only needs to sequence O(log 1) 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. 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 can be as high as (1) 2. In their celebrated proof that 2qfa's can recognize the non-regular language L = {a n b n | n > 0}, Kondacs and Watrous use a different probability amplification method, which yields machines with O((1) 2) states, and with runtime O(1 |w|), where w 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(|w|) time (i.e. the running time does not depend on the error bound) and which have O((1) 2 c) states for any given constant c > 1. Other methods, yielding machines whose state complexities are polylogarithmic in 1 , including one which halts in o(log(1)|w|) time, are also presented.