  The transformation of a molecular Hamiltonian from the fermionic space to the
qubit space results in a series of Pauli strings. Calculating the energy then
involves evaluating the expectation values of each of these strings, which
presents a significant bottleneck for applying variational quantum eigensolvers
(VQEs) in quantum chemistry. Unlike fermionic Hamiltonians, the terms in a
qubit Hamiltonian are additive. This work leverages this property to introduce
a novel method for extracting information from the partial qubit Hamiltonian,
thereby enhancing the efficiency of VQEs. This work introduces the SHARC-VQE
(Simplified Hamiltonian Approximation, Refinement, and Correction-VQE) method,
where the full molecular Hamiltonian is partitioned into two parts based on the
ease of quantum execution. The easy-to-execute part constitutes the Partial
Hamiltonian, and the remaining part, while more complex to execute, is
generally less significant. The latter is approximated by a refined operator
and added up as a correction into the partial Hamiltonian. SHARC-VQE
significantly reduces computational costs for molecular simulations. The cost
of a single energy measurement can be reduced from $O(\frac{N^4}{\epsilon^2})$
to $O(\frac{1}{\epsilon^2})$ for a system of $N$ qubits and accuracy
$\epsilon$, while the overall cost of VQE can be reduced from
$O(\frac{N^7}{\epsilon^2})$ to $O(\frac{N^3}{\epsilon^2})$. Furthermore,
measurement outcomes using SHARC-VQE are less prone to errors induced by noise
from quantum circuits, reducing the errors from 20-40% to 5-10% without any
additional error correction or mitigation technique. Additionally, the
SHARC-VQE is demonstrated as an initialization technique, where the simplified
partial Hamiltonian is used to identify an optimal starting point for a complex
problem.
