Nonlinear dynamics play a prominent role in many domains and are notoriously difficult to solve. Whereas previous quantum algorithms for general nonlinear equations have been severely limited due to the linearity of quantum mechanics, we gave the first efficient quantum algorithm for nonlinear differential equations with sufficiently strong dissipation. This is an exponential improvement over the best previous quantum algorithms, whose complexity is exponential in the evolution time. We also established a lower bound showing that nonlinear differential equations with sufficiently weak dissipation have worst-case complexity exponential in time, giving an almost tight classification of the quantum complexity of simulating nonlinear dynamics. Furthermore, we design the first quantum algorithm for training classical sparse neural networks with end-to-end settings.. We benchmark instances of training ResNet from 7 to 103 million parameters with sparse pruning applied to the Cifar-100 dataset, and we find that a quantum enhancement is possible at the early stage of learning. Our work shows that fault-tolerant quantum computing can contribute to the scalability and sustainability of most state-of-the-art, large-scale machine learning models.
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