  The underlying mechanism in the implementation of unitary operation on a
system with an external apparatus is studied. We implement the unitary time
evolution in the system as a physical phenomenon that results from the
interaction between the system and the apparatus. We investigate the
fundamental limitation of a good implementation for the desired unitary time
evolution. This limitation is manifested in the form of trade-off relations
between the accuracy of the implementation and quantum fluctuation of energy in
the external apparatus. Our relations clearly show that an accurate unitary
operation requires large energy fluctuation inside the apparatus originated
from quantum fluctuation.
