Eigenvector approximation leading to exponential speedup of quantum eigenvalue calculation.

@article{Jaksch2003EigenvectorAL,
  title={Eigenvector approximation leading to exponential speedup of quantum eigenvalue calculation.},
  author={Peter Jaksch and Anargyros Papageorgiou},
  journal={Physical review letters},
  year={2003},
  volume={91 25},
  pages={
          257902
        }
}
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 Schrödinger 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… 

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In fact, we can weaken this condition by considering the numerical error in solving the coarsely discretized problem. It suffices to assume that we have an approximation |Û