Resource-efficient energy-based operator selection in fermionic ADAPT-VQE via exact Hamiltonian transformation

The energy-based approach to operator selection in ADAPT-VQE relies on reconstructing the one-parameter energy landscape for each operator in the pool. In fermionic implementations, the cost of reconstructing this energy landscape often becomes a bottleneck. We address this issue through an exact Hamiltonian transformation that reformulates the one-parameter energy landscape according to a generator-dependent fragmentation of the transformed Hamiltonian. While our method is mathematically identical to standard fermionic Rotoselect, it effectively reduces its cost by about a factor of two, bringing it close to that of gradient-based ADAPT-VQE. We use this formulation to benchmark the gradient-based and energy-based selection approaches in combination with two ansatz-optimization strategies -- `last', where only the appended operator is optimized, or `full', where the full ansatz is re-optimized -- and with both fixed-orbital and orbital-optimized formulations. The benchmark comprises $\text{LiH}$, $\text{BeH}_2$, and $\text{H}_2\text{O}$ at both equilibrium and stretched geometries. In the weakly correlated regime, the `last' optimization strategy combined with energy-based selection enables the efficient construction of an accurate ansatz, while avoiding any VQE optimization. As correlation increases, full ansatz re-optimization and orbital optimization become the main factors governing convergence and overall resource cost. These results show that exact Hamiltonian transformations provide an effective route to reducing the measurement overhead of fermionic energy-based ADAPT-VQE. Moreover, the benchmark clarifies the relative role of operator scoring approach, re-optimization strategy, and orbital treatment in the performance of ADAPT-VQE.