  An algorithm that initializes a quantum register to a state with a specified
energy range is given, corresponding to a quantum implementation of the
celebrated Feit-Fleck method. This is performed by introducing a
nondeterministic quantum implementation of a standard spectral filtering
procedure combined with an apodization technique, allowing for accurate state
initialization. It is shown that the implementation requires only two ancilla
qubits. A lower bound for the total probability of success of this algorithm is
derived, showing that this scheme can be realized using a finite, relatively
low number of trials. Assuming the time evolution can be performed efficiently
and using a trial state polynomially close to the desired states, it is
demonstrated that the number of operations required scales polynomially with
the number of qubits. Tradeoffs between accuracy and performance are
demonstrated in a simple example: the harmonic oscillator. This algorithm would
be useful for the initialization phase of the simulation of quantum systems on
digital quantum computers.
