  In this work, we study probing for the more general problem of computing the
trace of a permutation of $A^{-1}$, say $PA^{-1}$. The motivation comes from
Lattice QCD where we need to construct "disconnected diagrams" to extract
flavor-separated Generalized Parton functions. In Lattice QCD, where the matrix
has a 4D toroidal lattice structure, these non-local operators correspond to a
$PA^{-1}$ where $P$ is the permutation relating to some displacement $\vec{p}$
in one or more dimensions. We focus on a single dimension displacement ($p$)
but our methods are general. We show that probing on $A^k$ or $(PA)^k$ do not
annihilate the largest magnitude elements. To resolve this issue, our
displacement-based probing works on $PA^k$ using a new coloring scheme that
works directly on appropriately displaced neighborhoods on the lattice. We
prove lower bounds on the number of colors needed, and study the effect of this
scheme on variance reduction, both theoretically and experimentally on a
real-world Lattice QCD calculation. We achieve orders of magnitude speedup over
the unprobed or the naively probed methods.
