  The efficient encoding of the fermionic Schr\"odinger equation as a spin
system Hamiltonian is a long-term problem. I describe an encoding for the
fermionic position space Schr\"odinger equation on a finite-volume periodic
lattice with a local potential. The challenging part of the construction is the
implementation of the kinetic energy operator, which is essentially the
Laplacian. The finite difference implementation on the lattice combines
contributions from neighboring lattice sites, which is complicated by fermionic
exchange symmetry.
  Two independently useful techniques developed here are operator filtering and
entanglement gadgets. Operator filtering is useful when a simple operator
acting on a subspace of the full Hilbert space has a desired set of
interactions. Occupation suppression of the complement of the subspace then
filters away unwanted contributions of the operator. Entanglement gadgets
encode the same information differently in two sets of qubits. We may then
independently choose the most efficient encoding for operators acting on the
qubits.
  The construction for the Laplacian described here has $\mathcal{O}\left(An
2^D\right)$ cost in bounded Pauli weight terms where $A$ is the number of
identical spinless fermions, $N=2^n$ is the number of lattice points in each
direction, and $D$ is the number of dimensions. The finite volume context
protects the gap between the ground state and the first excited state, yielding
polynomial time complexity with the box size.
