Quantum hardware demonstrations of relativistic calculations of molecular electric dipole moments: from light to heavy systems using Variational Quantum Eigensolver

The quantum-classical hybrid Variational Quantum Eigensolver (VQE) algorithm is recognized to be the method of choice to obtain ground state energies of quantum many-body systems in the noisy intermediate scale quantum (NISQ) era. This study not only extends the VQE algorithm to the relativistic regime, but also calculates a property other than energy, namely the molecular permanent electric dipole moment (PDM). We carry out 18-qubit quantum simulations to obtain ground state energies as well as PDMs of single-valence diatomic molecules, ranging from the light BeH to the heavy radioactive RaH molecule. We investigate the correlation trends in these systems as well as access the precision in our results. Furthermore, we measure the PDM of the moderately heavy SrH and SrF molecules on the optimized unitary coupled cluster state, using the state-of-the-art IonQ Aria-I quantum computer in an active space of 6 qubits. The associated quantum circuits for these computations were extensively optimized in view of limitations imposed by NISQ hardware. To that end, we employ an array of techniques, including the use of point group symmetries, integrating ZX-Calculus into our pipeline-based circuit optimization, and energy sort VQE procedure. Through these methods, we compress our 6-qubit quantum circuit from 280 two-qubit gates to 37 two-qubit gates (with a marginal trade-off of 0.33 and 0.31 percent in the PDM for SrH and SrF in their respective 6-spin orbital active spaces). We anticipate that our proof-of-concept demonstration lays the groundwork for future quantum hardware calculations involving heavy atoms and molecules.