Solving the max-3-cut problem using synchronized dissipative networks

Abstract

Many computational problems are intractable through classical computing and, as Moore's law is drawing to a halt, demand for finding alternative methods in tackling these problems is growing. Here, we propose a hardware implementable optical optimization heuristic for the NP-Hard max-3-cut problem based on a synchronized network of exciton-polariton condensates. The condensate network dynamical transients provide optically-fast annealing of the XY Hamiltonian to which we apply Goemans and Williamson random hyperplane technique, discretizing the XY ground state spin configuration to serve as ternary decision variables for an approximate optimal solution to the max-3-cut problem. Applications of our optical system are investigated in image segmentation tasks and constrained via minimization in circuit design, both showing good performance. Our work takes a step towards overcoming the limitation faced by Ising machines whose binary decision variables can only be mapped to the max-2-cut problem, whereas we realise an optical based XY machine to provide an approximate solution to the max-3-cut problem.

Minor embedding with Stuart-Landau oscillator networks

“…Recently, investigation has been devoted to the network dynamics of dissipatively coupled Stuart-Landau oscillators whose fixed point attractors correlate with minimal energy configurations of the XY Hamiltonian encoded in the network weights [10][11][12]. This opens up perspectives on analogue computing strategies based on oscillatory networks to heuristically solve a variety of complex graph problems such as the NP-hard max-3-cut problem [13], and the phase retrieval problem [14].…”

“…We demonstrate how the minor embedding technique realises a more experimentally friendly strategy to simulate and study graph problems with high degrees of connectivity in dissipatively coupled oscillatory systems. These include polariton condensates [13,17], photon condensates [18], non-degenerate optical parametric oscillators [19,20], and coupled microlaser arrays [21,22].…”