  The exploration of neural network quantum states has become widespread in the
search for ground states in complex quantum many-body systems. However,
achieving high precision remains challenging due to the intricate sign
structures and the exponential growth of the Hilbert space. In this work, we
propose a neural network state method that is confined to a significantly
smaller symmetric subspace, by the full space summation and Markov chain
Metropolis sampling approaches. Using symmetries and group theory, the proposed
method significantly reduces the number of parameters in neural network states
and achieves better accuracy and convergence properties. We validate our method
using the frustrated spin-$1/2$ $J_1$-$J_2$ antiferromagnetic Heisenberg chain
and compare its performance against NetKet, the standard library of neural
network states. The results indicate that our symmetrized neural network states
achieve a substantial improvement over the conventional neural network states
method, reducing energy errors by two orders of magnitude. We also compare
degenerate eigenstates with different quantum numbers, highlighting the
advantage of operating within a smaller variational space.
