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An extension of variational quantum eigensolver to calculate molecular orbitals

Ichio Kikuchi, Akihito Kikuchi

Published Jul 1, 2024• 0 citations
Publisher

Abstract

This article presents a construction of a variational quantum eigensolver that allows us to calculate molecular orbitals. As the authors of the presented work have discussed in their preceding works, the variables in the molecular Hartree-Fock equation are represented by a set of transformation matrices. Those matrices have common eigenvectors, and the corresponding eigenvalues give the individual values of the variables that construct the solution of the Hatree-Fock equation. In other words, one can get the wavefunctions and the orbital energies through eigenvalue problems, which could be evaluated by quantum phase estimation. In this article, we use this formalism for another purpose: the transformation matrices are used to construct the total energy function so that the minimization of the total energy leads to an alternative approach of variational quantum eigensolver if the computation is conducted in a quantum computer.

DOI
10.31219/osf.io/v7x9f

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This article presents a construction of a variational quantum eigensolver that allows us to calculate molecular orbitals. As the authors of the presented work have discussed in their preceding works, the variables in the molecular Hartree-Fock equation are represented by a set of transformation matrices. Those matrices have common eigenvectors, and the corresponding eigenvalues give the individual values of the variables that construct the solution of the Hatree-Fock equation. In other words, one can get the wavefunctions and the orbital energies through eigenvalue problems, which could be evaluated by quantum phase estimation. In this article, we use this formalism for another purpose: the transformation matrices are used to construct the total energy function so that the minimization of the total energy leads to an alternative approach of variational quantum eigensolver if the computation is conducted in a quantum computer.
In this article, we discuss how a kind of hybrid computation, which employs symbolic, numeric, classic, and quantum algorithms, allows us to conduct Hartree-Fock electronic structure computation of molecules. In the proposed algorithm, we replace the Hartree-Fock equations with a set of equations composed of multivariate polynomials. We transform those polynomials to the corresponding Gr\"obner bases, and then we investigate the corresponding quotient ring, wherein the orbital energies, the LCAO coefficients, or the atomic coordinates are represented by the variables in the ring. In this quotient ring, the variables generate the transformation matrices that represent the multiplication with the monomial bases, and the eigenvalues of those matrices compose the roots of the equation. The quantum phase estimation (QPE) algorithm enables us to record those roots in the quantum states, which would be used in the input data for more advanced and more accurate quantum computations.
Quantum computing is being extensively used in quantum chemistry, especially in simulating simple molecules and evaluating properties like the ground state energy, dipole moment, etc. The transformation of a molecular Hamiltonian from the fermionic space to the qubit space provides us with a series of Pauli strings and the energy calculation involves the evaluation of the expectation values of all these individual strings. This introduces a major bottleneck for applications of VQEs in quantum chemistry. Unlike the fermionic Hamiltonian, the terms in a qubit Hamiltonian are additive and the present paper exploits this property to describe a new approach for extracting information from the partial qubit Hamiltonian of simple molecules to design more efficient variational quantum eigensolvers. In the partial (qubit) Hamiltonian approach (PHA), the qubit Hamiltonian is studied term-by-term to understand their relative contributions to the overall energy, and a partial Hamiltonian is constructed with fewer Pauli strings that can resolve the entire Hamiltonian. With PHA, we can simulate molecules at a much lower computational cost with a truncated Hamiltonian. Additionally, the outcomes of the measurements with PHA quench the error due to noise introduced by the quantum circuits. We have also demonstrated the application of PHA as an initialization technique, where the simple partial Hamiltonian can be used to find an optimal starting point for a more complex system.
Near-term quantum computers will be limited in the number of qubits on which they can process information as well as the depth of the circuits that they can coherently carry out. To date, experimental demonstrations of algorithms such as the Variational Quantum Eigensolver (VQE) have been limited to small molecules using minimal basis sets for this reason. In this work we propose incorporating an orbital optimization scheme into quantum eigensolvers wherein a parametrized partial unitary transformation is applied to the basis functions set in order to reduce the number of qubits required for a given problem. The optimal transformation is found by minimizing the ground state energy with respect to this partial unitary matrix. Through numerical simulations of small molecules up to 16 spin orbitals, we demonstrate that this method has the ability to greatly extend the capabilities of near-term quantum computers with regard to the electronic structure problem. We find that VQE paired with orbital optimization consistently achieves lower ground state energies than traditional VQE when using the same number of qubits and even frequently achieves lower ground state energies than VQE methods using more qubits.
Recent advances in quantum technologies have led to the development of several quantum algorithms for computing molecular energetics. However, most existing approaches are limited to determining ground-state energies with relatively few studies addressing multilevel systems. In this work, we explore two quantum algorithms capable of addressing multilevel molecular orbital (MO) energetics: the subspace search variational quantum eigensolver (SSVQE) and iterative quantum phase estimation (IQPE). To benchmark their performance, we employed an exactly solvable Hamiltonian derived from the Hückel method. Both SSVQE and IQPE successfully reproduced the MO energies. We further discussed quantum circuit design and measurement noise using SSVQE. We found out the critical influence of quantum circuit design on computational accuracy. By examining SSVQE under noisy conditions, we could discuss its feasibility for implementation on near-term quantum hardware.
Classical simulation of molecular electronic structure is often limited by the rapid increase in computational cost associated with describing electron correlation. The Variational Quantum Eigensolver (VQE) is one hybrid quantum-classical approach ideally adapted to the Noisy Intermediate-Scale Quantum (NISQ) regime. This work applies the VQE algorithm to compute ground state energies of a set of molecules:H2,H3+,LiH,HF,OH−,H2O,BH3,CH4\mathrm{H}_{2}, \mathrm{H}_{3}^{+}, \text{LiH}, \text{HF}, \text{OH}^{-}, \mathrm{H}_{2} \mathrm{O}, \text{BH}_{3}, \text{CH}_{4}H2​,H3+​,LiH,HF,OH−,H2​O,BH3​,CH4​, andNH3\text{NH}_{3}NH3​. The approach utilizes the Unitary Coupled Cluster for Single and Double excitations (UCCSD) ansatz along with the Bravyi-Kitaev transformation for fermionic-to-qubit encoding. Quantum simulation outputs are systematically benchmarked against Full Configuration Interaction (FCI), a numerically exact traditional approach, and Unrestricted Hartree-Fock (UHF). The results establish a robust consistency of the VQE-obtained molecular ground state energies against standard FCI and UHF benchmarks. This research provides a critical benchmark, confirming that VQE has the capability for high-precision molecular simulations and playing the role of a pivotal stepping stone toward revolutionizing applications in drug discovery and materials science.
Determining the properties of molecules and materials is one of the premier applications of quantum computing. A major question in the field is how to use imperfect near-term quantum computers to solve problems of practical value. Inspired by the recently developed variants of the quantum counterpart of the equation-of-motion (qEOM) approach and the orbital-optimized variational quantum eigensolver (oo-VQE), we present a quantum algorithm (oo-VQE-qEOM) for the calculation of molecular properties by computing expectation values on a quantum computer. We perform noise-free quantum simulations of BeH 2 in the series of STO-3G/6-31G/6-31G* basis sets and of H 4 and H 2 O in 6-31G using an active space of four electrons and four spatial orbitals (8 qubits) to evaluate excitation energies, electronic absorption, and, for twisted H 4, circular dichroism spectra. We demonstrate that the proposed algorithm can reproduce the results of conventional classical CASSCF calculations for these molecular systems.