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arXiv:2112.15186 · Physical Review Research

Geometric quantum adiabatic methods for quantum chemistry

Hongye Yu, Deyu Lu, Qin Wu, Tzu-Chieh Wei

Published Jul 15, 2022• 4 citations
arXivHTMLPDF

Abstract

Quantum algorithms have been successfully applied in quantum chemistry to obtain the ground-state energy of small molecules. Although accurate near the equilibrium geometry, the results can become unreliable when the chemical bonds are broken at large distances. For any adiabatic approach, this is usually caused by serious issues, such as energy gap closing or level crossing along the adiabatic evolution path. In this work, we propose a quantum algorithm based on adiabatic evolution to obtain molecular eigenstates and eigenenergies in quantum chemistry, which exploits a smooth geometric deformation by continuously varying bond lengths and bond angles. We demonstrate its utility in several examples on a noiseless quantum simulator, including ${\mathrm{H}}_{2}\mathrm{O}, {\mathrm{CH}}_{2}$, and a chemical reaction of ${\mathrm{H}}_{2}+{\mathrm{D}}_{2}\ensuremath{\rightarrow}$ 2HD, by uniformly stretching chemical bonds. We find that this new algorithm solves the problems related to energy gap closing and level crossing along the adiabatic evolution path at large atomic distances. The new method performs more stably and achieves better accuracy than our previous adiabatic method [Yu and Wei, Phys. Rev. Research 3, 013104 (2021)]. Furthermore, our fidelity analysis demonstrates that even with finite bond length changes, our algorithm still achieves high fidelity for the ground state.

DOI
10.1103/physrevresearch.4.033045

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Quantum algorithms have been successfully applied in quantum chemistry to obtain the ground-state energy of small molecules. Although accurate near the equilibrium geometry, the results can become unreliable when the chemical bonds are broken at large distances. For any adiabatic approach, this is usually caused by serious issues, such as energy gap closing or level crossing along the adiabatic evolution path. In this work, we propose a quantum algorithm based on adiabatic evolution to obtain molecular eigenstates and eigenenergies in quantum chemistry, which exploits a smooth geometric deformation by continuously varying bond lengths and bond angles. We demonstrate its utility in several examples on a noiseless quantum simulator, including H2O,CH2{\mathrm{H}}_{2}\mathrm{O}, {\mathrm{CH}}_{2}H2​O,CH2​, and a chemical reaction of H2+D2\ensuremath→{\mathrm{H}}_{2}+{\mathrm{D}}_{2}\ensuremath{\rightarrow}H2​+D2​\ensuremath→ 2HD, by uniformly stretching chemical bonds. We find that this new algorithm solves the problems related to energy gap closing and level crossing along the adiabatic evolution path at large atomic distances. The new method performs more stably and achieves better accuracy than our previous adiabatic method [Yu and Wei, Phys. Rev. Research 3, 013104 (2021)]. Furthermore, our fidelity analysis demonstrates that even with finite bond length changes, our algorithm still achieves high fidelity for the ground state.
Abstract Modeling chemical reactions and complicated molecular systems has been proposed as the “killer application” of a future quantum computer. Accurate calculations of derivatives of molecular eigenenergies are essential toward this end, allowing for geometry optimization, transition state searches, predictions of the response to an applied electric or magnetic field, and molecular dynamics simulations. In this work, we survey methods to calculate energy derivatives, and present two new methods: one based on quantum phase estimation, the other on a low-order response approximation. We calculate asymptotic error bounds and approximate computational scalings for the methods presented. Implementing these methods, we perform geometry optimization on an experimental quantum processor, estimating the equilibrium bond length of the dihydrogen molecule to within 0.0140.0140.014 0.014 Å of the full configuration interaction value. Within the same experiment, we estimate the polarizability of the H 2{}_{2}2​ 2 molecule, finding agreement at the equilibrium bond length to within 0.060.060.06 0.06 a.u. ( 2%2 \%2% 2% relative error).
The search for new tasks related to quantum chemical calculations that can be performed on current quantum computers is important for the development of both quantum computing and quantum chemical calculations. Although chemical reaction calculations have a wide range of applications in quantum chemical calculations, there is no quantum algorithm for obtaining the activation energy (EaE_aEa​), which determines the rate of chemical reactions for any given substance. In this study, we propose a quantum algorithm for obtaining EaE_aEa​ for a given substance by optimizing the reaction path via molecular geometry generation using quantum circuits. The nudged elastic band method was used to optimize a reaction path, and the ground state calculation for each state on the reaction path involved either variational quantum eigensolver (VQE) or exact diagonalization (ED). The proposed algorithm was applied to H2 + H -> H + H2 reaction, and the reaction path was correctly optimized. Moreover, the activation energy was obtained with good accuracy regardless of whether VQE or ED is used for the ground state calculation. The results demonstrate the feasibility of performing the chemical reaction calculations using quantum algorithms.
Quantum computing promises to efficiently and accurately solve many important problems in quantum chemistry which elude classical solvers, such as the electronic structure problem of highly correlated materials. Two leading methods in solving the ground state problem are the Variational Quantum Eigensolver (VQE) and Adiabatic Quantum Computing (AQC) algorithms. VQE often struggles with convergence due to the energy landscape being highly non-convex and the existence of barren plateaux, and implementing AQC is beyond the capabilities of current quantum devices as it requires deep circuits. Adiabatically-inspired algorithms aim to fill this gap. In this paper, we first present a unifying framework for these algorithms and then benchmark the following methods: the Adiabatically Assisted VQE (AAVQE) (Garcia-Saez and Latorre (2018)), the Variational Adiabatic Quantum Computing (VAQC) (Harwood et al (2022)), and the Adiabatic Quantum Computing with Parametrized Quantum Circuits (AQC-PQC) (Kolotouros et al (2025)) algorithms. Second, we introduce a novel hybrid approach termed G-AQC-PQC, which generalizes the AQC-PQC method, and combines adiabatic-inspired initialization with the low-memory BFGS optimizer, reducing the quantum computational cost of the method. Third, we compare the accuracy of the methods for chemistry applications using the beryllium hydride molecule (BeH2_22​). We compare the approaches across a number of different choices (ansätze types, depth, discretization steps, initial Hamiltonian, adiabatic schedules and method used). Our results show that the G-AQC-PQC outperforms conventional VQE. We further discuss limitations such as the zero-gradient problem and identify regimes where adiabatically-inspired methods offer a tangible advantage for near-term quantum chemistry applications.
The search for new computational tasks of quantum chemistry that can be performed on current quantum computers is important for the development of quantum computing and quantum chemistry. Although calculations of chemical reactions have a wide range of applications in quantum chemical calculations, a quantum algorithm for obtaining activation energy EaE_aEa​, which determines the rate of chemical reactions, has not been performed. In this study, we propose a quantum algorithm for the chemical reaction path optimization to obtain EaE_aEa​. In our algorithm, quantum circuits can be used not only for the energy evaluation by the variational quantum eigensolver (VQE) but also for chemical reaction path generation. The chemical reaction path is obtained by encoding the initial reaction path to the circuit, operating parameterized gates, and extracting the path information by measurement. The nudged elastic band method was used for optimizing a reaction path, and the ground-state calculation for each state on the path was performed by the VQE or the exact diagonalization (ED). The proposed algorithm was applied to H2+H→H2+H\mathrm{H}_2 + \mathrm{H} \rightarrow \mathrm{H}_2 + \mathrm{H}H2​+H→H2​+H reaction, and we confirmed that EaE_aEa​ was obtained accurately in the case of both the VQE and the ED. We also obtained numerical results that the entanglement between the images accelerates the path optimization. From these results, we show the feasibility of performing fast and accurate chemical reaction calculations by using quantum algorithms.
Background: Solving nuclear many-body problems with an ab initio approach is widely recognized as a computationally challenging problem. Quantum computers offer a promising path to address this challenge. There are urgent needs to develop quantum algorithms for this purpose. Objective: In this work, we explore the application of the quantum algorithm of adiabatic state preparation with quantum phase estimation in ab initio nuclear structure theory. We focus on solving the low-lying spectra (including both the ground and excited states) of simple nuclear systems. Ideas: The efficiency of this algorithm is hindered by the emergence of small energy gaps (level crossings) during the adiabatic evolution. In order to improve the efficiency, we introduce techniques to avoid level crossings: 1) by suitable design of the reference Hamiltonian; 2) by insertions of perturbation terms to modify the adiabatic path. Results: We illustrate this algorithm by solving the deuteron ground state energy and the spectrum of the deuteron bounded in a harmonic oscillator trap implementing the IBM Qiskit quantum simulator. The quantum results agree well the classical results obtained by matrix diagonalization. Outlook: With our improvements to the efficiency, this algorithm provides a promising tool for investigating the low-lying spectra of complex nuclei on future quantum computers.
Classical algorithms for predicting the equilibrium geometry of strongly correlated molecules require expensive wave function methods that become impractical already for few-atom systems. In this work, we introduce a variational quantum algorithm for finding the most stable structure of a molecule by explicitly considering the parametric dependence of the electronic Hamiltonian on the nuclear coordinates. The equilibrium geometry of the molecule is obtained by minimizing a more general cost function that depends on both the quantum circuit and the Hamiltonian parameters, which are simultaneously optimized at each step. The algorithm is applied to find the equilibrium geometries of the H2,{\mathrm{H}}_{2},H2​, H3+,{\mathrm{H}}_{3}{}^{+},H3​+, BeH2,{\mathrm{BeH}}_{2},BeH2​, and H2O{\mathrm{H}}_{2}\mathrm{O}H2​O molecules. The quantum circuits used to prepare the electronic ground state for each molecule were designed using an adaptive algorithm where excitation gates in the form of Givens rotations are selected according to the norm of their gradient. All quantum simulations are performed using the PennyLane library for quantum differentiable programming. The optimized geometrical parameters for the simulated molecules show an excellent agreement with their counterparts computed using classical quantum chemistry methods.