  Higher-order interactions in spin-based Hamiltonians are crucial in
addressing numerous fundamentally significant physical problems. In this work,
Rydberg-atom graph gadgets are introduced to effectively program $K$-th order
interactions within a Rydberg atom system. This approach facilitates the
determination of the ground states of an Ising-type Hamiltonian, encoded to
solve higher-order unconstrained optimization problems. A favorable scaling
behavior, $O(N^K)$, is expected in terms of the number of atoms required for
$N$-vertex hypergraph optimization problems.
