Quantum algorithms for image classification and nonlinear differential equations

This thesis presents two works related to quantum algorithms.In the first part, we propose a new quantum neural network for image classification, which is able to classify the parity of the MNIST dataset with full resolution with a test accuracy of up to 97.5% without any classical pre-processing or post-processing.Our architecture is based on a mixture of experts whose model function is the sum of the model functions of each expert.We encode the input with amplitude encoding, which allows us to encode full-resolution MNIST images with 10 qubits and to implement a convolution on the whole image with just a single one- qubit gate.Our training algorithm is based on training all the experts together, which significantly improves trainability with respect to training each expert independently. In fact, in the limit of infinitely many experts, our training algorithm can perfectly fit the training data. Our results demonstrate the potential of our quantum neural network to achieve high-accuracy image classification with minimal quantum resources, paving the way for more scalable and efficient quantum machine learning models.In the second part, we study with classical simulations and analytical calculations the quantum algorithm of [Lloyd emph{et al.}, arXiv:2011.06571] for solving nonlinear differential equations, showing its strong limitations.