Some results on equivalence of multi-letter quantum finite automata

Two quantum finite automata are equivalent if for all input string ω over the input alphabet the two automata accept ω with equal probability. In [Theoret. Comput. Sci. 410 (2009) 3006-3017], it was shown that a k1-letter QFA A1 and a k2-letter QFA A2 over Σ = {σ}, are equivalent if and only if they are (n1 + n2) 4 + k − 1-equivalent where ni is the number of states of Ai, i = 1, 2, and k = max{k1, k2}. In this letter, we improve the above upper-bound to (n1 + n 2 2 − 1) + k. This also answers an open problem of Qiu et al. [Acta Informatica 48 (2011) 271-290]. Further, we show that, in the case of Σ = {σ1, · · · , σt} with 2 ≤ t <∞, there exists an integer z such that A1 and A2 are equivalent if and only if they satisfy z-equivalent.