  Given an initial quantum state |psi_I> and a final quantum state |psi_F> in a
Hilbert space, there exist Hamiltonians H under which |psi_I> evolves into
|psi_F>. Consider the following quantum brachistochrone problem: Subject to the
constraint that the difference between the largest and smallest eigenvalues of
H is held fixed, which H achieves this transformation in the least 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. This is because for such Hamiltonians the path from |psi_I> to
|psi_F> can be made 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.
