Determining the equivalence for one-way quantum finite automata

Two quantum finite automata are equivalent if for any input string x the two automata accept x with equal probability. In this paper, we first focus on determining the equivalence for one-way quantum finite automata with control language (CL-1QFAs) defined by Bertoni et al., and then, as an application, we address the equivalence problem for measure-many one-way quantum finite automata (MM-1QFAs) introduced by Kondacs and Watrous. More specifically, we obtain that: (i)Two CL-1QFAs A"1 and A"2 with control languages (regular languages) L"1 and L"2, respectively, are equivalent if and only if they are (c"1n"1^2+c"2n"2^2-1)-equivalent, where n"1 and n"2 are the numbers of states in A"1 and A"2, respectively, and c"1 and c"2 are the numbers of states in the minimal DFAs that recognize L"1 and L"2, respectively. Furthermore, if L"1 and L"2 are given in the form of DFAs, with m"1 and m"2 states, respectively, then there exists a polynomial-time algorithm running in time O((m"1n"1^2+m"2n"2^2)^4) that takes as input A"1 and A"2 and determines whether they are equivalent. (ii)(As an application of item (i)): Two MM-1QFAs A"1 and A"2 with n"1 and n"2 states, respectively, are equivalent if and only if they are (3n"1^2+3n"2^2-1)-equivalent. Furthermore, there is a polynomial-time algorithm running in time O((3n"1^2+3n"2^2)^4) that takes as input A"1 and A"2 and determines whether A"1 and A"2 are equivalent.

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