Super-Exponential Size Advantage of Quantum Finite Automata with Mixed States

Quantum finite automata with mixed states are proved to be super-exponentially more concise rather than quantum finite automata with pure states. It was proved earlier by A.Ambainis and R.Freivalds that quantum finite automata with pure states can have exponentially smaller number of states than deterministic finite automata recognizing the same language. There was a never published ”folk theorem” proving that quantum finite automata with mixed states are no more than superexponentially more concise than deterministic finite automata. It was not known whether the super-exponential advantage of quantum automata is really achievable. We use a novel proof technique based on Kolmogorov complexity to prove that there is an infinite sequence of distinct integers n such that there are languages Ln in a 4-letter alphabet such that there are quantum finite automata with mixed states with 2n + 1 states recognizing the language Ln with probability 4 while any deterministic finite automaton recognizing Ln needs to have at least e states.