Hamiltonian-Guided Leverage Embedding: Robust Subspace Compression for Efficient QAOA Parameter Estimation

The Quantum Approximate Optimization Algorithm (QAOA) is a hybrid quantum-classical framework for combinatorial optimization on near-term quantum devices. A central bottleneck is the classical estimation of its variational parameters {\gamma} and {\beta}, which must be optimized over a high-dimensional, non-convex landscape corrupted by sampling noise. We observe that the classical feature matrices constructed from QAOA measurement samples exhibit pronounced low-rank structure, and exploit this property for noise-robust, reduced-dimension parameter search. We present the Hamiltonian-Guided Leverage Embedding (HGLE) algorithm - a hybrid pipeline that encodes low-energy quantum samples into a weighted Ising feature matrix and compresses it via leverage-score row sampling, provably preserving the dominant rank-rsubspace geometry. The compressed representation drives a classical trust-region loop for ({\gamma}, {\beta}) estimation at a fraction of the original cost. We provide formal guarantees for rank preservation and energy approximation error, and demonstrate robustness across problem types (Max-Cut, Maximum Independent Set) and graph topologies of varying density.