  Grover's algorithm is a quantum search algorithm that proceeds by repeated
applications of the Grover operator and the Oracle until the state evolves to
one of the target states. In the standard version of the algorithm, the Grover
operator inverts the sign on only one state. Here we provide an exact solution
to the problem of performing Grover's search where the Grover operator inverts
the sign on N states. We show the underlying structure in terms of the
eigenspectrum of the generalized Hamiltonian, and derive an appropriate initial
state to perform the Grover evolution. This allows us to use the quantum phase
estimation algorithm to solve the search problem in this generalized case,
completely bypassing the Grover algorithm altogether. We obtain a time
complexity of this case of $\sqrt(D/M^\alpha)$ where D is the search space
dimension, M is the number of target states, and $\alpha$ ~ 1, which is close
to the optimal scaling.
