We find the optimal universal way of manipulating a single qubit, \psi(thet
a, phi)], such that (theta, phi)-->(theta-alpha, phi -beta). Such optimal t
ransformations fall into two classes. For 0 less than or equal to alpha les
s than or equal to pi /2, the optimal map is the identity and the fidelity
varies monotonically from 1 (for alpha =0) to 1/2 (for alpha=pi /2). For pi
/2s less than or equal to alpha less than or equal to pi, the optimal map
is the universal-NOT gate and the fidelity varies monotonically from 1/2 (f
or alpha=pi /2) to 2/3 (for alpha=pi). The fidelity 2/3 is equal to the fid
elity of measurement. It is therefore rather surprising that for some value
s of alpha the fidelity is lower than 2/3. For instance, a universal square
root of NOT operation is more difficult to approximate than the universal
NOT gate itself.