Dimensionality Reduction of QAOA Parameter Space with Kernel PCA for Max-Cut

Abstract

The Quantum Approximate Optimization Algorithm (QAOA) is a leading variational algorithm for combinatorial optimization on near term quantum devices. As circuit depth increases, the number of optimization parameters grows, making the search landscape increasingly nonlinear and difficult to optimize. Previous studies have shown that optimal QAOA parameters often lie on a low dimensional manifold that can be approximated using Principal Component Analysis (PCA) at shallow circuit depths. However, the effectiveness of PCA decreases at higher depths because the underlying parameter manifold becomes increasingly nonlinear. In this work, we investigate Kernel Principal Component Analysis (KPCA) with a radial basis function kernel as a nonlinear dimensionality reduction technique for QAOA parameter optimization. The model is trained using 200 graphs from each of 3 graph families, namely Erdos-Renyi, Barabasi-Albert, and Watts-Strogatz, with graph sizes ranging from 7 to 10 nodes. Performance is evaluated on 30 test graphs containing 12 nodes at circuit depths 1, 2, 4, and 8. Experimental results demonstrate that KPCA consistently outperforms PCA at deeper circuit depths across all graph families. At depth 8, KPCA achieves approximation ratios above 0.86, while PCA declines to approximately 0.81 to 0.83. Both methods reduce the number of quantum circuit evaluations by more than 93 percent relative to unrestricted QAOA optimization. These findings suggest that nonlinear kernel methods more effectively capture the structure of the QAOA parameter manifold and provide a practical approach for scaling variational quantum optimization to deeper circuits.