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In this work, we study probing for the more general problem of computing thetrace of a permutation of $A^{-1}$, say $PA^{-1}$. The motivation comes fromLattice QCD where we need to construct "disconnected diagrams" to extractflavor-separated Generalized Parton functions. In Lattice QCD, where the matrixhas 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 notannihilate the largest magnitude elements. To resolve this issue, ourdisplacement-based probing works on $PA^k$ using a new coloring scheme thatworks directly on appropriately displaced neighborhoods on the lattice. Weprove lower bounds on the number of colors needed, and study the effect of thisscheme on variance reduction, both theoretically and experimentally on areal-world Lattice QCD calculation. We achieve orders of magnitude speedup overthe unprobed or the naively probed methods.

Probing for the Trace Estimation of a Permuted Matrix Inverse Corresponding to a Lattice Displacement