  In recent years there has been a growing interest in the physical
implementation of classical spin models through networks of optical
oscillators. However, a key missing step in this mapping is to formally prove
that the dynamics of such a nonlinear dynamical system is toward minimizing a
global cost function which is equivalent with the spin model Hamiltonian. Here,
we introduce a minimal dynamical model for a network of dissipatively coupled
optical oscillators and prove that the dynamics of such a system is governed by
a Lyapunov function that serves as a cost function for the system. This cost
function is in general a function of both phases and intensities of the
oscillators and depends strongly on the pump parameter. In case of bipartite
network topologies, the amplitudes of the oscillators become identical in the
steady state and the cost function reduces to the XY Hamiltonian. In the
general case for non-trivial network topologies, however, the cost function
approaches the XY Hamiltonian only in the strong pump limit. We show that by
adiabatically tuning the pump parameter, the network can largely avoid trapping
into the local minima of the governing cost function and stabilize into the
ground state of the associated XY Hamiltonian. These results show the great
potential of laser networks for unconventional computing.
