A quantum clock must satisfy two basic constraints. The first is a bound on
the time resolution of the clock given by the difference between its maxim
um and minimum energy eigenvalues. The second follows from Holevo's bound o
n how much classical information can be encoded in a quantum system. We sho
w that asymptotically, as the dimension of the Hilbert space of the clock t
ends to infinity, both constraints can be satisfied simultaneously. The exp
erimental realization of such an optimal quantum clock using trapped ions i
s discussed.