  Cold atoms in an optical cavity have been widely used for quantum simulations
of many-body physics, where the quantum control capability has been advancing
rapidly in recent years. Here, we show the atom cavity system is universal for
quantum optimization with arbitrary connectivity. We consider a single-mode
cavity and develop a Raman coupling scheme by which the engineered quantum
Hamiltonian for atoms directly encodes number partition problems (NPPs). The
programmability is introduced by placing the atoms at different positions in
the cavity with optical tweezers. The NPP solution is encoded in the ground
state of atomic qubits coupled through a photonic cavity mode, that can be
reached by adiabatic quantum computing (AQC). We construct an explicit mapping
for the 3-SAT and vertex cover problems to be efficiently encoded by the cavity
system, which costs linear overhead in the number of atomic qubits. The atom
cavity encoding is further extended to quadratic unconstrained binary
optimization (QUBO) problems. The encoding protocol is optimal in the cost of
atom number scaling with the number of binary degrees of freedom of the
computation problem. Our theory implies the atom cavity system is a promising
quantum optimization platform searching for practical quantum advantage.
