  We reformulate the projected imaginary-time evolution of Full Configuration
Interaction Quantum Monte Carlo in terms of a Lagrangian minimization. This
naturally leads to the admission of polynomial complex wavefunction
parameterizations, circumventing the exponential scaling of the approach. While
previously these functions have traditionally inhabited the domain of
Variational Monte Carlo, we consider recently developments for the
identification of deep-learning neural networks to optimize this Lagrangian,
which can be written as a modification of the propagator for the wavefunction
dynamics. We demonstrate this approach with a form of Tensor Network State, and
use it to find solutions to the strongly-correlated Hubbard model, as well as
its application to a fully periodic ab-initio Graphene sheet. The number of
variables which can be simultaneously optimized greatly exceeds alternative
formulations of Variational Monte Carlo, allowing for systematic improvability
of the wavefunction flexibility towards exactness for a number of different
forms, whilst blurring the line between traditional Variational and Projector
quantum Monte Carlo approaches.
