  Quantum annealing is a promising approach to heuristically solving difficult
combinatorial optimization problems. However, the connectivity limitations in
current devices lead to an exponential degradation of performance on general
problems. We propose an architecture for a quantum annealer that achieves full
connectivity and full programmability while using a number of physical
resources only linear in the number of spins. We do so by application of
carefully engineered periodic modulations of oscillator-based qubits, resulting
in a Floquet Hamiltonian in which all the interactions are tunable; this
flexibility comes at a cost of the coupling strengths between spins being
smaller than they would be had the spins been directly coupled. Our proposal is
well-suited to implementation with superconducting circuits, and we give
analytical and numerical evidence that fully connected, fully programmable
quantum annealers with $1000$ qubits could be constructed with Josephson
parametric oscillators having cavity-photon lifetimes of $\sim 100$
microseconds, and other system-parameter values that are routinely achieved
with current technology. Our approach could also have impact beyond quantum
annealing, since it readily extends to bosonic quantum simulators and would
allow the study of models with arbitrary connectivity between lattice sites.
