Quantum simulation of Cayley-tree Ising Hamiltonians with three-dimensional Rydberg atoms

Significant efforts are being directed toward developing a quantum simulator capable of solving combinatorial optimization problems. The challenges are Hamiltonian programming in terms of high-dimensional qubit connectivities and large-scale implementations. Here, we report a quantum simulation demonstration of Ising Hamiltonians with up to N = 22 spins mapped on various Cayley-tree graphs. For this, we use three-dimensional arrangements of Rydberg single atoms arranged in such a way that their Rydberg atoms and blockaded strong couplings respectively represent the vertices 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 many-body ground states and Neel's order formations are experimentally probed. The antiferromagnetic phase in regular Cayley trees and frustrated competing ground states in a dual-center Cayley tree are directly observed, demonstrating the possibilities of high-dimensional qubit connections in quantum simulators.