Hierarchical Discovery of Adiabatic Hamiltonian Paths and RL Schedules for Quantum Linear System Solvers

Quantum linear-system solvers can provide asymptotic speedups under structured access as- sumptions, but finite-size performance depends strongly on the spectral encoding, Hamiltonian path, traversal schedule, and implementation model. We study a hierarchical framework for the quantum linear systems problem (QLSP) that sep- arates admissible Hamiltonian-path design from adaptive traversal control. The outer loop searches constrained deformations of analytically valid adi- abatic backbones within implementable opera- tor libraries, while the inner loop is restricted to residual schedule control around a derivative- aware local-adiabatic prior. This residual class can be optimized by derivative-free search, dif- ferentiable optimal control, or family-wise RL; the present experiments instantiate derivative- free residual search and frame RL as the nat- ural amortized-control extension rather than as unconstrained Hamiltonian invention. In this sense, the framework acts as a two-stage auto- matic algorithm-design pipeline: the outer layer automatically adapts valid path families, while the inner layer automatically refines residual traversal schedules. We do not seek to improve worst-case asymptotic query complexity beyond optimal- scaling QLSP solvers; instead, the framework tar- gets engineering-relevant finite-size fidelity–time tradeoffs, family-dependent adaptation, and diag- nostic transparency. Simulations show the clearest gains on structured gap-amplified and precondi- tioned families. Ancilla-assisted extensions can improve gap structure, subspace anchoring, and preconditioning, but do not evade known optimal- scaling query-complexity limits.