  Artificial neural networks have been established as a universally applicable
tool in both science and industry. In the context of symmetric open quantum
systems, specifically the restricted Boltzmann machine architecture has emerged
as a natural and highly compressed representation of the density matrix for
open spin-$1/2$ quantum systems via a direct mapping of spins to the visible
neuron layer. Here we present a bit encoding scheme for a highly efficient and
scalable representation of bosonic Fock number states in the restricted
Boltzmann machine neural network architecture, extending its applicability
beyond pure spin-$1/2$ open quantum systems. In contrast to common density
matrix implementations, the complexity of the neural network scales only with
the number of bit-encoded neurons rather than the maximum boson number.
Crucially, in the high occupation regime its information compression efficiency
is shown to surpass even maximally optimized density matrix implementations,
where a projector method is used to access the sparsest Hilbert space
representation available.
