SHARC-VQE: Simplified Hamiltonian approach with refinement and correction enabled variational quantum eigensolver for molecular simulation

Quantum computing is finding increasingly more applications in quantum chemistry, particularly to simulate electronic structure and molecular properties of simple systems. 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(N4ϵ2) to O(1ϵ2) for a system of N qubits and accuracy ϵ, while the overall cost of VQE can be reduced from O(N7ϵ2) to O(N3ϵ2). 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. In addition, 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. Overall, this method improves the efficiency of VQEs and enhances the accuracy and reliability of quantum simulations by mitigating noise and overcoming computational challenges.