Relativistic variational-quantum-eigensolver calculations of molecular electric dipole moments on quantum hardware

The quantum--classical hybrid variational quantum eigensolver (VQE) algorithm is among the most actively studied topics in atomic and molecular calculations on quantum computers, yet few studies address properties other than energies or account for relativistic effects. This work presents high-precision 18-qubit relativistic VQE simulations for calculating the permanent electric dipole moments (PDMs) of alkaline-earth metal monohydride molecules (BeH to RaH) on traditional computers, and 6- and 12-qubit PDM computations for SrH on IonQ quantum devices. To achieve high precision on current noisy intermediate-scale era quantum hardware, we apply various resource-reduction methods, including reinforcement learning-based and causal flow preserving ZX--calculus routines, along with error mitigation and postselection techniques. Our approach reduces the two-qubit gate count in our 12-qubit circuit by $99.71%$, with only a $2.35%$ trade-off in precision for PDM when evaluated classically within a suitably chosen active space. On the current generation IonQ Forte-I hardware, the error in PDM is $\ensuremath{-}1.17%$ relative to classical calculations and only $1.21%$ compared with the unoptimized circuit.