Efficient probability amplification in two-way quantum finite automata

Abstract

In classical computation, one only needs to sequence O ( log 1 ϵ ) 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 ϵ . 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 ϵ can be as high as ( 1 ϵ ) 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 ϵ ) 2 ) states, and with runtime O ( 1 ϵ | 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 ϵ ) 2 c ) states for any given constant c > 1 . Other methods, yielding machines whose state complexities are polylogarithmic in 1 ϵ , including one which halts in o ( log ( 1 ϵ ) | w | ) time, are also presented.