I'm trying to understand the problem of state preparation for quantum phase estimation (QPE). Specifically how states are prepared adiabatically.
I have a couple of questions:
1). Typically when one thinks of adiabatic state preparation, annealing comes to mind. However, it is also possible to do time evolution on a gate-based quantum computer with a time-dependent Hamiltonian in order to adiabatically prepare a state. Is the latter option also considered in the context of adiabatic state preparation for QPE (the few papers I have found talk about annealing)?
2). In practice we do not expect to prepare $|\psi\rangle = |\psi_0\rangle$ but instead a superposition of states that has some non-negligible overlap with the ground state. Does this mean that we can relax the requirement of adiabaticity and instead prepare a superposition of the ground and some excited states? If so, would this help to avoid the time taken for adiabatic state preparation scaling as $\frac{1}{\Delta^2}$?
Any pointers to references are also much appreciated.
