  We propose a photonic quantum simulator for anti-ferromagnetic spin systems
based on reservoir engineering. We consider a scheme where quadratically driven
dissipative Kerr cavities are indirectly coupled via lossy ancillary cavities.
We show that the ancillary cavities can produce an effective dissipative and
Hamiltonian anti-ferromagnetic-like coupling between the cavities. By solving
the master equation for a triangular cavity configuration, we demonstrate that
the non-equilibrium steady state of the system bears full analogy with the
ground state of an antiferromagnetic Ising model, exhibiting key signatures of
frustration. We show that when the effective photon hopping amplitude is zero,
the engineered non-local dissipation alone is capable of inducing
antiferromagnetic interaction and frustration. This simple scheme can be
generalised to arbitrary lattice geometries, providing a fully controllable
recipe for simulating antiferromagnetism and frustration on a controlled
quantum optical platform.
