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We investigate quantum annealing (QA) via the transverse interaction with XY-type anisotropy to the ground state problems for a toy model system composed of 4 S=l/2 quantum spins interacting with anti-ferromagnetic interactions. By solving the Schrodinger equation for the toy model system, we find that the QA succeeds in deriving the ground state of the toy model system, if the target system does not have non-trivial degeneracy in the ground state. Then, we also And that the QA selectively obtains a desirable solution among the degenerate ground states by tuning the XY-type anisotropy appropriately, when the target system has non-trivial degeneracy in the ground state. Then, the spin wave theory also finds that the toy model indicates similar properties in an infinite-volume limit. Next, we find that the ground state of the target system can be obtained even if we start annealing procedure from states including excited states of the kinetic energy term, when we set the time interval to be large in solving the Schrodinger equation.