  In this article we present a full description of the quantum Kerr Ising
model---a linear optical network of parametrically pumped Kerr non-linearities.
We consider the non-dissapative Kerr Ising model and, using variational
techniques, show that the energy spectrum is primarily determined by the
adjacency matrix in the Ising model and exhibits highly non-classical cat like
eigenstates. We then introduce dissipation to give a quantum mechanical
treatment of the measurement process based on homodyne detection via the
conditional stochastic Schrodinger equation. Finally we identify a quantum
advantage in comparison to the classical analogue for the example of two
anti-ferromagnetic cavities.
