  The learning process for multi layered neural networks with many nodes makes
heavy demands on computational resources. In some neural network models, the
learning formulas, such as the Widrow-Hoff formula, do not change the
eigenvectors of the weight matrix while flatting the eigenvalues. In infinity,
this iterative formulas result in terms formed by the principal components of
the weight matrix: i.e., the eigenvectors corresponding to the non-zero
eigenvalues. In quantum computing, the phase estimation algorithm is known to
provide speed-ups over the conventional algorithms for the eigenvalue-related
problems. Combining the quantum amplitude amplification with the phase
estimation algorithm, a quantum implementation model for artificial neural
networks using the Widrow-Hoff learning rule is presented. The complexity of
the model is found to be linear in the size of the weight matrix. This provides
a quadratic improvement over the classical algorithms.
