Quantum many-body systems provide a rich framework for exploring and understanding the fundamental laws of physics. By studying the collective behavior and emergent phenomena that arise from the intricate microscopic interactions among particles, we can deepen our understanding of quantum mechanics and gain insight into its broader implications for macroscopic observations. However, quantum many-body systems pose significant computational challenges, as the information contained in the many-body wave function grows exponentially with the size of the system. This exponential scaling, coupled with the presence of strong interparticle correlations, makes the accurate description of these systems difficult, if not impossible, for traditional analytical or perturbative techniques. In this interdisciplinary approach, we aim to solve the quantum many-body problem by representing the trial wave function of a variational Monte Carlo calculation by a so-called neural-network quantum state. These states, as their name suggests, are rooted in artificial neural networks and serve as a novel alternative to conventional parameterizations of the trial wave function. In addition to reviewing key concepts in quantum many-body theory and machine learning, we investigate a diverse set of continuous-space systems with varying levels of complexity. Starting from a pedagogical overview of an exactly solvable system of bosons in one dimension, we work our way up to strongly-interacting fermionic systems in three dimensions, including dilute neutron matter and ultra-cold Fermi gases. The highly non-perturbative interactions featured in these systems motivate the development of innovative neural-network quantum states, capable of discovering strong correlations while maintaining required symmetries and boundary conditions. We accompany this study with the description of two distinct implementations of neural-network quantum states, each with their unique goals and strategies. Our findings indicate that neural-network quantum states provide a powerful and flexible strategy for investigating a wide range of quantum phenomena, without relying on prior assumptions about the underlying physics as in traditional approaches.
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