Penalty-Free Two-Step Optimization of Higher-Order Ising Problems for Two-Dimensional Line-Controlled RIS

Reconfigurable intelligent surfaces (RISs) are often assumed to allow continuous phase control over all elements, leading to hardware cost that scales with the number of elements. Treating the phase of each element as a discrete variable is essential for improving cost effectiveness toward ubiquitous RIS deployment. However, the resulting discrete optimization problem is inherently difficult to solve. To address this challenge, this letter proposes a two-dimensional line-control method to reduce the degrees of freedom of the phase variables. The formulation yields a fourth-order objective function and is not directly compatible with physical optimizers such as coherent Ising machines and quantum annealers, which are designed for quadratic interactions. Conventional methods for reducing the order of the objective function with additional auxiliary variables increase the number of variables and require additional penalty parameters, limiting scalability. We therefore propose a two-step optimization method that transforms the fourth-order objective into two successive quadratic optimization problems. For a RIS with 5,476 elements, the required number of discrete variables is reduced from 11,100 to 5,476. Experiments using a real coherent Ising machine demonstrated that the proposed approach solved the discrete-phase optimization problem with 5,476 elements, while limiting the beamforming-gain loss to 2 dB compared with the full continuous-control case.