Imaginary time quantum evolution as a Riemannian gradient flow

We identify imaginary time quantum evolution as a Riemannian gradient flow on the unitary group. We show that with an appropriately chosen step size, the discretized gradient flow and imaginary time evolution are the same up to a controllable error. We go on to use this observation to develop quantum algorithms that implement imaginary time evolution on quantum computers. And finally, derive performance bounds for adaptive quantum algorithms that prepare the ground state of a desired Hamiltonian.