Training-free operator selection for quantum circuit construction from a heterogeneous-graph representation

Abstract

Automated construction of variational quantum circuits remains a major bottleneck in quantum chemistry, largely because adaptive VQE methods repeatedly screen large operator pools using quantum-gradient criteria. For a canonical Hartree--Fock reference and normalized double-excitation generators, the initial ADAPT gradient magnitude is exactly the corresponding antisymmetrized two-electron Hamiltonian coupling. We encode this static signal as an excitation-edge feature of a heterogeneous graph, enabling a training-free classical ranking without current-state candidate-gradient measurements. Across statevector benchmarks from eight to twenty qubits, incremental construction reaches chemical accuracy with operator counts close to full gradient-based selection while avoiding repeated quantum screening. Supervised and residual models are used as controls to test whether learned corrections improve this low-cost rule. Static learned readouts provide no consistent held-out advantage, whereas state-conditioned features improve selected teacher-forced ranking metrics but require additional information and do not establish a closed-loop reduction in operator count. As a complementary higher-rank contribution, we derive an exact shared-residual formulation that reduces Hamiltonian applications for triple- and quadruple-excitation screening from $2C$ or $C+1$ to one while preserving all scores and rankings. Results on stretched H$_6$ and N$_2$ benchmarks show that these higher-rank operators can repair a singles-and-doubles expressivity limit.