**Faster than Hermitian quantum mechanics**

**Abstract:**

Given an initial quantum state |*I*>
and a final quantum state |*F*> in a Hilbert space, there exist Hamiltonians *H* under which |*I*> evolves into
|*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 τ?
For Hermitian Hamiltonians τ has a nonzero lower bound. However, among
non-Hermitian PT-symmetric Hamiltonians satisfying the same energy constraint,
τ can be made arbitrarily small without violating the time-energy uncertainty principle. This is because for such Hamiltonians the path from |I> to |*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.