  Despite rapid progress in the development of quantum algorithms in quantum
computing as well as numerical simulation methods in classical computing for
atomic and molecular applications, no systematic and comprehensive electronic
structure study of atomic systems that covers almost all of the elements in the
periodic table using a single quantum algorithm has been reported. In this
work, we address this gap by implementing the recently-proposed quantum
algorithm, the Bayesian Phase Difference Estimation (BPDE) approach, to compute
accurately fine-structure splittings, which are relativistic in origin and it
also depends on quantum many-body (electron correlation) effects, of
appropriately chosen states of atomic systems, including highly-charged
superheavy ions. Our numerical simulations reveal that the BPDE algorithm, in
the Dirac--Coulomb--Breit framework, can predict the fine-structure splitting
of Boron-like ions to within 605.3 cm$^{-1}$ of root mean square deviations
from the experimental ones, in the (1s, 2s, 2p, 3s, 3p) active space. We
performed our simulations of relativistic and electron correlation effects on
Graphics Processing Unit (GPU) by utilizing NVIDIA's cuQuantum, and observe a
$\times 42.7$ speedup as compared to the CPU-only simulations in an 18-qubit
active space.
