  An algorithm is presented for approximating arbitrary powers of a black box
unitary operation, $\mathcal{U}^t$, where $t$ is a real number, and
$\mathcal{U}$ is a black box implementing an unknown unitary. The complexity of
this algorithm is calculated in terms of the number of calls to the black box,
the errors in the approximation, and a certain `gap' parameter. For general
$\mathcal{U}$ and large $t$, one should apply $\mathcal{U}$ a total of $\lfloor
t \rfloor$ times followed by our procedure for approximating the fractional
power $\mathcal{U}^{t-\lfloor t \rfloor}$. An example is also given where for
large integers $t$ this method is more efficient than direct application of $t$
copies of $\mathcal{U}$. Further applications and related algorithms are also
discussed.
