  We consider the Feynman-Kitaev formalism applied to a spin chain described by
the transverse field Ising model. This formalism consists of building a
Hamiltonian whose ground state encodes the time evolution of the spin chain at
discrete time steps. To find this ground state, variational wave functions
parameterised by artificial neural networks -- also known as neural quantum
states (NQSs) -- are used. Our work focuses on assessing, in the context of the
Feynman-Kitaev formalism, two properties of NQSs: expressivity (the possibility
that variational parameters can be set to values such that the NQS is faithful
to the true ground state of the system) and trainability (the process of
reaching said values). We find that the considered NQSs are capable of
accurately approximating the true ground state of the system, i.e., they are
expressive enough ans\"atze. However, extensive hyperparameter tuning
experiments show that, empirically, reaching the set of values for the
variational parameters that correctly describe the ground state becomes ever
more difficult as the number of time steps increase because the true ground
state becomes more entangled, and the probability distribution starts to spread
across the Hilbert space canonical basis.
