  Quantum many-body problems are some of the most challenging problems in
science and are central to demystifying some exotic quantum phenomena, e.g.,
high-temperature superconductors. The combination of neural networks (NN) for
representing quantum states, coupled with the Variational Monte Carlo (VMC)
algorithm, has been shown to be a promising method for solving such problems.
However, the run-time of this approach scales quadratically with the number of
simulated particles, constraining the practically usable NN to - in machine
learning terms - minuscule sizes (<10M parameters). Considering the many
breakthroughs brought by extreme NN in the +1B parameters scale to other
domains, lifting this constraint could significantly expand the set of quantum
systems we can accurately simulate on classical computers, both in size and
complexity. We propose a NN architecture called Vector-Quantized Neural Quantum
States (VQ-NQS) that utilizes vector-quantization techniques to leverage
redundancies in the local-energy calculations of the VMC algorithm - the source
of the quadratic scaling. In our preliminary experiments, we demonstrate VQ-NQS
ability to reproduce the ground state of the 2D Heisenberg model across various
system sizes, while reporting a significant reduction of about ${\times}10$ in
the number of FLOPs in the local-energy calculation.
