Hierarchy and equivalence of multi-letter quantum finite automata

Multi-letter quantum finite automata (QFAs) were a new one-way QFA model proposed recently by Belovs, Rosmanis, and Smotrovs [5], and they showed that multi-letter QFAs can precisely accept some regular languages ((a+b)∗b) that are unacceptable by the usual one-way QFAs. In this paper, we continue to deal with multi-letter QFAs, and we mainly focus on two issues: (1) we show that (k+1)-letter QFAs are computationally more powerful than k-letter QFAs, that is, (k + 1)-letter QFAs can accept some regular languages unacceptable by any k-letter QFA. A comparison to the usual one-way QFAs is made by some examples; (2) we give a polynomial-time algorithm to determine whether any given two multi-letter QFAs for accepting unary languages are equivalent, and, indeed, we prove that a k1-letter QFA A1 and another k2-letter QFA A2 are equivalent if and only if they are (n1 +n2) + k− 1-equivalent, where n1 and n2 are the numbers of states of A1 and A2, respectively, and k = max(k1, k2). This method, generalized appropriately, may apply to dealing with more general cases. Some issues are addressed for further consideration.