  Neural-network state representations of quantum many-body systems are
attracting great attention and more rigorous quantitative analysis about their
expressibility and complexity is warranted. Our analysis of the restricted
Boltzmann machine (RBM) state representation of one-dimensional (1D) quantum
spin systems provides new insight into their computational complexity. We
define a class of long-range-fast-decay (LRFD) RBM states with quantifiable
upper bounds on truncation errors and provide numerical evidence for a large
class of 1D quantum systems that may be approximated by LRFD RBMs of at most
polynomial complexities. These results lead us to conjecture that the ground
states of a wide range of quantum systems may be exactly represented by LRFD
RBMs or a variant of them, even in cases where other state representations
become less efficient. At last, we provide the relations between multiple
typical state manifolds. Our work proposes a paradigm for doing complexity
analysis for generic long-range RBMs which naturally yields a further
classification of this manifold. This paradigm and our characterization of
their nonlocal structures may pave the way for understanding the natural
measure of complexity for quantum many-body states described by RBMs and are
generalizable for higher-dimensional systems and deep neural-network quantum
states.
