  Through the introduction of auxiliary fermions, or an enlarged spin space,
one can map local fermion Hamiltonians onto local spin Hamiltonians, at the
expense of introducing a set of additional constraints. We present a
variational Monte-Carlo framework to study fermionic systems through
higher-dimensional (>1D) Jordan-Wigner transformations. We provide exact
solutions to the parity and Gauss-law constraints that are encountered in
bosonization procedures. We study the $t$-$V$ model in 2D and demonstrate how
both the ground state and the low-energy excitation spectra can be retrieved in
combination with neural network quantum state ansatze.
