Quantum annealing of Cayley-tree Ising spins at small scales

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.