The learning process for multilayered 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, these iterative formulas result in terms formed by the principal components of the weight matrix, namely, the eigenvectors corresponding to the non-zero eigenvalues. In quantum computing, the phase

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... estimation algorithm is known to provide speedups 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.Quanta 2018; 7: 7–18.