  Significant efforts are being directed towards developing a quantum annealer
capable of solving combinatorial optimization problems. The challenges are
Hamiltonian programming and large-scale implementations. Here we report quantum
annealing demonstration of Ising Hamiltonians programmed with up to $N=22$
spins mapped on various Cayley tree graphs. Experiments are performed with a
Rydberg-atom quantum simulator, in which rubidium single atoms are arranged in
three dimensional space in such a way that their Rydberg atoms and blockaded
strong couplings respectively represent the nodes and edges of each graph.
Three different Cayley-tree graphs of $Z=3$ neighbors and of up to $S=4$ shells
are constructed, and their ground-state phases and N\'{e}el's order formations
are probed. In good agreement with model calculations, the anti-ferromagnetic
phase in regular Cayley trees and frustrated competing ground-states in a
dual-center Cayley tree are directly observed. This demonstrates the
possibilities of high-dimensional qubit connection programming in quantum
simulators.
