This thesis presents the development and application of novel Neural Network Quantum States (NQS) for the simulation of strongly correlated quantum matter in continuous space. We augment traditional Slater-Jastrow-Backflow ansaetze with modern machine learning architectures, such as permutation-invariant DeepSets and permutation-equivariant graph neural networks (GNNs), dubbed MP-NQS, to construct compact, flexible and highly expressive variational models for both bosonic and fermionic systems.

A key innovation of this work is the incorporation of periodic boundary conditions into the network design of the NQS, allowing the accurate simulation of condensed matter systems and materials and the computation of thermodynamic properties from first principles. We demonstrate the effectiveness of this approach by simulating the phase diagrams of benchmark systems, including $^4$He in one and two dimensions and the homogeneous electron gas in three dimensions, capturing superfluidity, crystallization, and other emergent phenomena.

For fermionic systems, we further enhance the variational ansatz by integrating the Pfaffian determinant as anti-symmetric prior, allowing us to describe pairing correlations in ultracold Fermi gases across the BCSâ BEC crossover. In addition to bulk applications, we extend the framework to molecular systems, demonstrate its applicability to small molecules, and compute real-time dynamical properties using the time-dependent variational principle. Furthermore, we introduce an algorithm to access their finite-temperature properties and nuclear quantum effects through variational (path-integral) molecular dynamics.

Overall, the methods introduced in this work significantly broaden the applicability of variational quantum Monte Carlo by combining physical priors with the flexibility of deep learning. They pave the way for accurate, scalable, and transferable modeling of quantum matter, with potential impact on quantum chemistry, materials science, and strongly correlated condensed matter systems.