Finite cyclic quantum state machines: a topological perspective
Allen Parks · Journal of Physics A Mathematical and General · 2001
Finite cyclic quantum state machines (FCQSMs) are characterized by free actions of finite cyclic groups upon odd-dimensional spheres. This provides for a covering space representation for all such machines. FCQSM simulation, as well as simple, quotient, and split FCQSMs are defined within this context. The notions of dynamical and process symmetries for FCQSMs are introduced and it is shown that although all FCQSMs obey a weak version of a FCQSM symmetry principle, only those which adhere to a strong version of this principle can simulate special FCQSMs defined by their dynamical symmetries. Finally, a simulation complexity index and an induced topological complexity index are defined for FCQSMs and an order relation is developed in terms of these indices which serves as a more precise statement of the weak version of the FCQSM symmetry principle. These indices are also shown to be related to FCQSMs which do not conform to the strong version of the FCQSM symmetry principle.