Abstract: Quantum computing has attracted extensive attention due to its intrinsic parallel computation and physical realization. Quantum computing model is one of the most important problems in the field of quantum computing. Sequential quantum machine and quantum sequential machine are two important quantum computing models, and they are essentially equivalent. In this paper, we study the properties of uninitialized sequential quantum machine by utilizing algebraic methods, which provide a theoretical basis for the study of sequential quantum machine. Firstly, we introduce the definition of homomorphism of uninitialized sequ-ential quantum machines. Some homomorphic properties of uninitialized sequential quantum machines are obtained, and the homomorphism theorem of uninitialized sequential quantum machines is proved. Secondly, we study the congruence properties on the set of input-output pairs of uninitialized sequential quantum machine and the matrix algebra properties of uninitialized sequential quantum machines. Moreover, a commutative uninitialized sequential quantum machine is defined, and its properties are discussed. Finally, the equivalence of uninitialized sequential quantum machines is established, and the equivalence of two initial vectors of a commutative uninitialized sequential quantum machine is discussed. The obtained results improve some existing results.

Key words: sequential quantum machine, quantum sequential machine, homomorphism, congruence, equivalence, commutativity