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Extracting Information from Partial Hamiltonian of Simple Molecules for Designing Efficient Variational Quantum Eigensolvers

Harshdeep Singh, Sabyashachi Mishra, Sonjoy Majumder

Published Jan 1, 2023• 0 citations
Publisher

Abstract

This work aims to study the qubit Hamiltonian of a molecule term-by-term to understand their contributions, and if any information can be extracted by just considering a few selected terms of the Hamiltonian, a technique called partial Hamiltonian analysis (PHA).
DOI
10.1364/quantum.2023.qtu3a.18

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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.
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