Pushing deep neural quantum states toward machine precision

Accessing theoretically the ground state of interacting quantum matter has remained a notorious challenge, especially for complex two-dimensional systems. Recent developments have highlighted the potential of neural quantum states to solve the quantum many-body problem by encoding the quantum many-body wave function into artificial neural networks. So far, however, this method faces the critical limitation that the training of modern large-scale deep network architectures has not yet been possible, thereby failing to capitalize on the full power of artificial neural networks. Here, we introduce a minimum-step stochastic reconfiguration (MinSR) optimization algorithm, which allows us to train unprecedentedly deep neural quantum states with up to one million parameters and 64 layers. We demonstrate our method in the paradigmatic spin-1/2 Heisenberg J1-J2 models on the square and the triangular lattices. In these systems, the deep neural networks approach different levels of machine precision on modern GPU and TPU hardware and yield significantly better variational energies compared to existing variational results. The accurate numerical results suggest the existence of gapless spin liquid in the most frustrated regime of the corresponding models. This opens up a new stage where the neural quantum state is not only applied for benchmark purposes but also helps to deepen the understanding of controversial quantum many-body systems.