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Mathematics

Efficient Ground State Energy Estimation of LiCoO2 Using the FMO-VQE Hybrid Quantum Algorithm

Yoonho Choe, Doyeon Kim, Doha Kim, Younghun Kwon

Published Dec 22, 2025• 1 citation
Publisher

Abstract

The Variational Quantum Eigensolver (VQE) is a quantum algorithm for estimating ground-state energies, with promising applications in material science, drug discovery, and battery research. A key challenge is the limited number of qubits available on current quantum devices, which restricts the size of molecular systems that can be studied. To address this limitation, we apply the Fragment Molecular Orbital (FMO) method in combination with VQE, referred to as FMO-VQE. This approach divides a system into smaller fragments, making the quantum calculations more tractable. While earlier studies demonstrated this method only for hydrogen clusters, we extend the application to lithium cobalt oxide, a widely used cathode material in lithium-ion batteries. Using FMO-VQE, we estimate the ground-state energy of this complex system while reducing the number of required qubits from 24 to 14, without significant loss of accuracy compared to classical methods. This reduction highlights the potential of FMO-VQE to overcome hardware limitations and make quantum simulations of larger molecules feasible. The results suggest a practical path for applying near-term quantum computers to real-world challenges, opening opportunities for advancements in the battery industry and drug design.

DOI
10.3390/math14010044

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The Variational Quantum Eigensolver (VQE) is a quantum algorithm for estimating ground-state energies, with promising applications in material science, drug discovery, and battery research. A key challenge is the limited number of qubits available on current quantum devices, which restricts the size of molecular systems that can be studied. To address this limitation, we apply the Fragment Molecular Orbital (FMO) method in combination with VQE, referred to as FMO-VQE. This approach divides a system into smaller fragments, making the quantum calculations more tractable. While earlier studies demonstrated this method only for hydrogen clusters, we extend the application to lithium cobalt oxide, a widely used cathode material in lithium-ion batteries. Using FMO-VQE, we estimate the ground-state energy of this complex system while reducing the number of required qubits from 24 to 14, without significant loss of accuracy compared to classical methods. This reduction highlights the potential of FMO-VQE to overcome hardware limitations and make quantum simulations of larger molecules feasible. The results suggest a practical path for applying near-term quantum computers to real-world challenges, opening opportunities for advancements in the battery industry and drug design.
Quantum computers offer significant potential for complex system analysis, yet their application in large systems is hindered by limitations such as qubit availability and quantum hardware noise. While the variational quantum eigensolver (VQE) was proposed to address these issues, its scalability remains limited. Many efforts, including new ansätze and Hamiltonian modifications, have been made to overcome these challenges. In this work, we introduced the novel Fragment Molecular Orbital/Variational Quantum Eigensolver (FMO/VQE) algorithm. This method combines the fragment molecular orbital (FMO) approach with VQE and efficiently utilizes qubits for quantum chemistry simulations. Employing the UCCSD ansatz, the FMO/VQE achieved an absolute error of just 0.053 mHa with 8 qubits in a [Formula: see text] system using the STO-3G basis set, and an error of 1.376 mHa with 16 qubits in a [Formula: see text] system with the 6-31G basis set. These results indicated a significant advancement in scalability over conventional VQE, maintaining accuracy with fewer qubits. Therefore, our FMO/VQE method exemplifies how integrating fragment-based quantum chemistry with quantum algorithms can enhance scalability, facilitating more complex molecular simulations and aligning with quantum computing advancements.
Abstract Material discovery is a phenomenon practiced since the evolution of the world. The discovery of materials has led to significant development in varied fields such as Science, Engineering and Technology. Computationally simulating molecules has been an area of interest in the industry as well as academia. However, simulating large molecules can be computationally expensive in terms of computing power and complexity. Quantum computing is a recent development that can improve the efficiency in predicting properties of atoms and molecules which will be useful for material design. The Variational Quantum Eigensolver (VQE) is one such quantum algorithm used to calculate the ground state energy of molecules or ions. In this study, we have done a comparative analysis of the parameters that constitute the VQE algorithm. This includes components such as basis, qubit mapping, ansatz, and optimizers used. We have also developed a database consisting of 79 single atoms and their variations of oxidation states and 33 molecules with the data of their Hamiltonian and ground state energy and dipole moment.
Exponential scaling presents a significant challenge in electronic structure calculations performed on classical computers. This paper explores how quantum computer algorithms can accurately represent quantum systems. Variational Quantum Eigensolver (VQE) algorithm is used to compute the ground state energy of hydrogen and helium sequences by implementing variational principle and quantum gates as trial wavefunction. This technique combines classical optimization with quantum computing calculations to simulate quantum systems on noisy and resource-limited computers. The resulting calculated energy is highly consistent to the corresponding exact values and Hartree-Fock calculations with a trend of when the number of atoms increases the calculated energy becomes more negative, leading to a decrease in the percentage error. Moreover, the convergence of the ground state energy of hydrogen and helium atoms was effectively optimized. The desired energy was reached, proven by adjusting the expectation value, and gradually achieving unity in state overlap. These findings demonstrate the VQE method's accuracy in calculating simple quantum systems and its scalability for larger atomic and molecular system, such as those in quantum chemistry and material science. However, challenges in quantum computer simulations, such as limited in qubit numbers and the presence of noise, require further advancements. Therefore, implementing a larger basis sets, advanced qubit mapping, specific chemistry ansatz, and flexible optimization techniques is one way to improve overall calculation.
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.
Accurate prediction of binding energies between drug candidates and target molecules is essential in computational drug discovery, as stronger (more negative) binding energies suggest higher efficacy. This energy corresponds to the ground state of the molecular system and can be computed using methods such as Full Configuration Interaction (FCI), Hartree-Fock (HF), Variational Quantum Eigensolver (VQE), and Quantum Phase Estimation (QPE). While FCI offers benchmark accuracy, its computational cost renders it impractical for larger systems. HF scales efficiently but at the expense of precision. QPE, though theoretically robust, is currently constrained by hardware limitations. VQE emerges as a practical alternative for Noisy Intermediate-Scale Quantum (NISQ) devices, balancing accuracy and feasibility. Nevertheless, quantum noise significantly affects VQE performance. To address this, we incorporate Zero Noise Extrapolation (ZNE) to mitigate errors and improve energy estimates. This study compares these approaches and demonstrates the efficacy of the VQEZNE framework in generating reliable binding energy calculations for drug design applications.
Variational quantum eigensolver (VQE) is a hybrid quantum-classical technique that leverages noisy intermediate scale quantum (NISQ) hardware to obtain the minimum eigenvalue of a model Hamiltonian. VQE has so far been used to simulate condensed matter systems as well as quantum chemistry of small molecules. In this work, we employ VQE methods to obtain the ground-state energy of LiCoO2_22​, a candidate transition metal oxide used for battery cathodes. We simulate Li2_22​Co2_22​O4_44​ and Co2_22​O4_44​ gas-phase models, which represent the lithiated and delithiated states during the discharge and the charge of the Li-ion battery, respectively. Computations are performed using a statevector simulator with a single reference state for three different trial wavefunctions: unitary coupled-cluster singles and doubles (UCCSD), unitary coupled-cluster generalized singles and doubles (UCCGSD) and k-unitary pair coupled-cluster generalized singles and doubles (k-UpCCGSD). The resources in terms of circuit depth, two-qubit entangling gates and wavefunction parameters are analyzed. We find that the k-UpCCGSD with k=5 produces results similar to UCCSD but at a lower cost. Finally, the performance of VQE methods is benchmarked against the classical wavefunction-based methods, such as coupled-cluster singles and doubles (CCSD) and complete active space configuration interaction (CASCI). Our results show that VQE methods quantitatively agree with the results obtained from CCSD. However, the comparison against the CASCI results clearly suggests that advanced trial wavefunctions are likely necessary to capture the multi-reference characteristics as well as the correlations emerging from high-level electronic excitations.