Nature provides us with a restricted set of microscopic interactions. The q
uestion is whether we can synthesize out of these fundamental interactions
an arbitrary unitary operator. In this paper we present a constructive algo
rithm for realization of any unitary operator which acts on a (truncated) H
ilbert space of a single bosonic mode. The algorithm itself is not unitary
because it involves a conditional measurement. However, it does yield a con
stant probability of the conditional measurement which does not depend on t
he input state of the bosonic system. We consider a physical implementation
of unitary transformations acting on one-dimensional vibrational states of
a trapped ion. As an example we present an algorithm which realizes the di
screte Fourier transform.