CIMs lift combinatorial optimization problems into **continuous model spaces**.

While canonical Ising problems are formulated in terms of binary (spin) variables, the degenerate optical parametric oscillator (DOPO) modes that represent CIM states are fundamentally analog in nature. To map the dynamics of a network of coupled DOPOs to an Ising optimization problem, CIMs exploit the “discretizing” pitchfork bifurcation that DOPOs exhibit at pump threshold. Below threshold, early in the pump schedule of a CIM computation, the DOPO network can be conceived as a sort of “soft spin” Ising model. It remains an open theoretical question how the CIM strategy of embedding Ising optimization in a model space with both continuous and discrete regions may give rise to advantages or disadvantages for physical optimization dynamics — especially when we admit the possibility of coherent quantum dynamics and entanglement in DOPO networks. To add further complexity, we may consider the idea of using device configurations closely related to DOPOs (such as non-degenerate optical parametric oscillators) to implement networks of coupled XY-type degrees-of-freedom. Future hardware systems could support smooth, scheduled transitions among soft-spin, binary Ising and XY regions of model space. Could clever strategies be formulated to avoid computational bottlenecks?

**See also:**

T. Leleu* et al.*, “Destabilization of Local Minima in Analog Spin Systems by Correction of Amplitude Heterogeneity,” Phys. Rev. Lett. **122**, 040607 (2019).

R. Hamerly and H. Mabuchi, “Optical Devices Based on Limit Cycles and Amplification in Semiconductor Optical Cavities,” Phys. Rev. Applied **4**, 024016 (2016).

S. Gopalakrishnan *et al.*, “Exploring models of associative memory via cavity quantum electrodynamics,” Phil. Mag. **92**, 353 (2012).

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