  The interest in decoherence-free, or noiseless subsystems (DFS/NSs) of
quantum systems is both of fundamental and practical interest. Understanding
the invariance of a set of states under certain transformations is mutually
associated with a better understanding of some fundamental aspects of quantum
mechanics as well as the practical utility of invariant subsystems. For
example, DFS/NSs are potentially useful for protecting quantum information in
quantum cryptography and quantum computing as well as enabling universal
computation. Here we discuss transformations which are compatible with a DFS/NS
that is composed of d-state systems which protect against collective noise.
They are compatible in the sense that they do not take the logical (encoded)
states outside of the DFS/NS during the transformation. Furthermore, it is
shown that the Hamiltonian evolutions derived here can be used to perform
universal quantum computation on a three qudit DFS/NS. Many of the methods used
in our derivations are directly applicable to a large variety of DFS/NSs. More
generally, we may also state that these transformations are compatible with
collective motions.
