In the paper we consider measured-once (MO-QFA) oneway quantum finite automaton. We prove that for MO-QFA Q that (1/2+ε)-accepts (ε ∈ (0,1/2)) regular language L it holds that dim(Q) = Ω (log dim (A)/log log dim (A)). In the case ε ∈ (3/8, 1/2) we have more precise lower bound dim(Q) = Ω (log dim (A)) where A is a minimal deterministic finite automaton accepting L, dim(Q), and dim(A) are complexity (number of states) of automata Q and A respectively, (1/2 - ε) is the error of Q.