Permutation Groups and the Strength of Quantum Finite Automata with Mixed States

It was proved earlier by A. Ambainis and R. Freivalds that the quantum finite automata with pure states can have exponentially smaller number of states than the deterministic finite automata recognizing the same language. There is a never published “folk theorem” claiming that the quantum finite automata with mixed states are no more than super-exponentially concise than the deterministic finite automata. It is not known whether the super-exponential advantage of the quantum automata with mixed states is really achievable. We show how this problem can be reduced to a certain problem about permutation groups, namely: (1) if there is a fixed constant c and an infinite sequence of distinct integers n such that for each n there is a group Gn of permutations of the set {1, 2, . . . , n} such that |Gn| = e Ω(n log n) and the pairwise Hamming distance of permutations is at least c · n, then (2) there is an infinite sequence of languages Ln such that for each language there is a quantum finite automata with mixed states that recognizes the language Ln and has O(n) states, while any deterministic finite automaton recognizing Ln must have at least e Ω(n log n) states. We do not know whether (1) is true, but we provide a list of several known results on permutation groups, that possibly could be used to prove (1). However, note that if (1) turns out to be false, it does not imply, that (2) is also false.