We consider several problems that involve finding the eigenvalues and
generating the eigenstates of unknown unitary gates. We first examine
Controlled-U gates that act on qubits, and assume that we know the eigenvalues.
It is then shown how to use singlet states to produce qubits in the eigenstates
of the gate. We then remove the assumption that we know the eigenvalues and
show how to both find the eigenvalues and produce qubits in the eigenstates.
Finally, we look at the case where the unitary operator acts on qutrits and has
eigenvalues of 1 and -1, where the eigenvalue 1 is doubly degenerate. The
eigenstates are unknown. We are able to use a singlet state to produce a qutrit
in the eigenstate corresponding to the -1 eigenvalue.Comment: Latex, 10 pages, no figure
