Learning Symmetry-Preserving Sampling Circuits for Sample-Based Quantum Diagonalization

Learning quantum sampling circuits requires searching discrete architectures under physical constraints and a downstream scientific objective. We study this problem in sample-based quantum diagonalization (SQD), which estimates molecular ground-state energies by classically diagonalizing a Hamiltonian in a subspace sampled from a quantum circuit. A widely used sampler is the local unitary cluster Jastrow (LUCJ) ansatz parameterized from classical coupled-cluster amplitudes, which can become unreliable under strong correlation. We formulate sampler construction as a sequential decision process and introduce a neuro-symbolic reinforcement-learning approach that combines an actor–critic policy with a particle-number-preserving gate vocabulary and symbolic construction rules. A terminal reward based on the SQD energy and circuit length directly guides the search toward accurate, compact samplers. Across the tested molecular geometries of N₂, H₂O, and LiH, shallow searched circuits recover active-space ground-state energies in simulation. In the larger N₂ (14-electron, 10-orbital) active space, a searched circuit achieves chemical accuracy with 48 transpiled CZ gates, compared with 31,260 in the automatically constructed LUCJ baseline. Ablations attribute the subspace-coverage advantage to the symmetry-preserving search space, while the learned policy improves search consistency. Executions on IBM Heron-class hardware retain 65–74% of measurements in the valid particle-number sector at equilibrium geometries. The studied active spaces remain small enough that recovered energies do not discriminate subspace quality after configuration recovery. These results establish circuit efficiency and hardware feasibility for learned SQD samplers, while leaving their advantage in subspace quality at larger scales an open question.