SHARC-VQE: Simplified Hamiltonian Approach with Refinement and Correction enabled Variational Quantum Eigensolver for Molecular Simulation

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.