  Neural network architectures have been increasingly used to represent quantum
many-body wave functions. In this context, a large number of parameters is
necessary, causing a crisis in traditional optimization methods. The Stochastic
Reconfiguration (SR) approach has been widely used in the past for cases with a
limited number of parameters $P$, since it involves the inversion of a $P
\times P$ matrix and becomes computationally infeasible when $P$ exceeds a few
thousands. This is the major limitation in the context of deep learning, where
the number of parameters significantly surpasses the number of samples $M$ used
for stochastic estimates ($P \gg M$). Here, we show that SR can be applied
exactly by leveraging a simple linear algebra identity to reduce the problem to
the inversion of a manageable $M \times M$ matrix. We demonstrate the
effectiveness of our method by optimizing a Deep Transformer architecture
featuring approximately 300,000 parameters without any assumption on the sign
structure of the ground state. Our approach achieves state-of-the-art
ground-state energy on the $J_1$-$J_2$ Heisenberg model in the challenging
point $J_2/J_1=0.5$ on the $10\times10$ square lattice, a demanding benchmark
problem in the field of highly-frustrated magnetism. This work marks a
significant step forward in the scalability and efficiency of SR for Neural
Network Quantum States (NNQS), making them a promising method to investigate
unknown phases of matter of quantum systems, where other methods struggle.

Strongly Correlated Electrons;Disordered Systems and Neural Networks