Coherent-cluster-state generation in networks of degenerate optical parametric oscillators

Cluster states are versatile quantum resources and an essential building block for measurement-based quantum computing. The possibility to generate cluster states in specific systems may thus serve as an indicator if and to what extent these systems can be harnessed for quantum technologies and quantum information processing in particular. Here, we apply this analysis to networks of degenerate optical parametric oscillators (DOPOs), also called coherent Ising machines (CIMs). CIMs are distinguished by their highly flexible coupling capabilities, which makes it possible to use them, e.g., to emulate large spin systems. As CIMs typically operate with coherent states (and superpositions thereof), it is natural to consider cluster states formed by superpositions of coherent states, i.e., coherent cluster states. As we show, such coherent cluster states can, under ideal conditions, be generated in DOPO networks with the help of beam splitters and classical pumps. Our subsequent numerical analysis provides the minimum requirements for the generation of coherent cluster states under realistic conditions. Moreover, we discuss how nonequilibrium pumps can improve the generation of coherent cluster states. In order to assess the quality of the cluster state generation, we map the generated states to an effective spin space using modular variables, which allows us to apply entanglement criteria tailored for spin-based cluster states.