We present an efficient method for preparing the initial state required by
the eigenvalue approximation quantum algorithm of Abrams and Lloyd. Our method
can be applied when solving continuous Hermitian eigenproblems, e.g., the
Schroedinger equation, on a discrete grid. We start with a classically obtained
eigenvector for a problem discretized on a coarse grid, and we efficiently
construct, quantum mechanically, an approximation of the same eigenvector on a
fine grid. We use this approximation as the initial state for the eigenvalue
estimation algorithm, and show the relationship between its success probability
and the size of the coarse grid.Comment: 4 page
