  For any pair of quantum states, an initial state |I> and a final quantum
state |F>, in a Hilbert space, there are many Hamiltonians H under which |I>
evolves into |F>. Let us impose the constraint that the difference between the
largest and smallest eigenvalues of H, E_max and E_min, is held fixed. We can
then determine the Hamiltonian H that satisfies this constraint and achieves
the transformation from the initial state to the final state in the least
possible time \tau. For Hermitian Hamiltonians, \tau has a nonzero lower bound.
However, among non-Hermitian PT-symmetric Hamiltonians satisfying the same
energy constraint, \tau can be made arbitrarily small without violating the
time-energy uncertainty principle. The minimum value of \tau can be made
arbitrarily small because for PT-symmetric Hamiltonians the path from the
vector |I> to the vector |F>, as measured using the Hilbert-space metric
appropriate for this theory, can be made arbitrarily short. The mechanism
described here is similar to that in general relativity in which the distance
between two space-time points can be made small if they are connected by a
wormhole. This result may have applications in quantum computing.
