  We study the virtual Euler characteristics of sheaves over Quot schemes of
curves, establishing that these invariants fit into a topological quantum field
theory (TQFT) valued in $\mathbb{Z}[[q]]$. Utilizing Quot scheme
compactifications alongside the TQFT framework, we derive presentations of the
small quantum $K$-ring of the Grassmannian. Our approach offers a new method
for finding explicit formulas for quantum $K$-invariants.
