Quantum machine learning has the potential for broad industrial applications, and the development of quantum algorithms for improving the performance of neural networks is of particular interest given the central role they play in machine learning today. In this paper we present quantum algorithms for training and evaluating feedforward neural networks based on the canonical classical feedforward and backpropagation algorithms. Our algorithms rely on an efficient quantum subroutine for approximating the inner products between vectors, and on storing intermediate values in quantum random access memory for fast retrieval at later stages. The running times of our algorithms can be quadratically faster than their classical counterparts, since they depend linearly on the number of neurons in the network, as opposed to the number of edges as in the classical case. This makes our algorithms suited for large-scale, highly-connected networks where the number of edges in the network dominates the classical algorithmic running time.
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