Freivalds and Smith [R. Freivalds, C.H. Smith Memory limited inductive inference machines, Springer Lecture Notes in Computer Science 621 (1992) 19-29] proved that probabilistic limited memory inductive inference machines can learn with probability 1 certain classes of total recursive functions, which cannot be learned by deterministic limited memory inductive inference machines. We introduce quantum limited memory inductive inference machines as quantum finite automata acting as inductive inference machines. These machines, we show, can learn classes of total recursive functions not learnable by any deterministic, nor even by probabilistic, limited memory inductive inference machines.