Qa${\text{s}}^2$QD: Learning to design circuits for scalable sample-based quantum diagonalization

Ground-state energy estimation (GSE) is one of the most promising applications of quantum computers for understanding quantum systems across physics, chemistry, and materials science, yet remains challenging at large scales. Recent advances in sample-based quantum diagonalization (SQD) offer a promising route to this task. However, its scalability critically depends on the sampling circuit used to construct the reduced subspace. To address this bottleneck, here we first reformulate GSE tasks as a family-level learning problem across related Hamiltonians, a setting that naturally arises in many-body physics and quantum chemistry. Building on this formulation, we develop Qa${\text{s}}^2$QD, a learning-guided quantum architecture search framework for automatically designing effective SQD sampling circuits. At its core is a multimodal predictor that jointly learns from circuit structure, Hamiltonian information, and low-shot observations to predict high-shot SQD performance, enabling circuit-performance knowledge to transfer across related Hamiltonians. We systematically evaluate Qa${\text{s}}^2$QD on quantum systems with up to $40$ qubits. On TFIM and XXZ benchmarks, the multimodal predictor preserves strong circuit-performance rankings using only $1\%$ of the reference shots per circuit, enabling Qa${\text{s}}^2$QD to reduce the error by up to $91.4\%$ over vanilla SQD. These results demonstrate the potential of Qa${\text{s}}^2$QD for tackling challenging quantum systems and advancing quantum computing toward practical utility.