The Complexity of Probabilistic versus Quantum Finite Automata

Abstract

We present a language $L_n$ which is recognizable by a probabilistic finite automaton (PFA) with probability $1 - ε$ for all $ε> 0$ with $O(log^2n)$ states, with a deterministic finite automaton (DFA) with O(n) states, but a quantum finite automaton (QFA) needs at least $2^{Ω(n/ \log n)}$ states.

6 pages, 2 figures