Feynmans ideas to employ quantum systems for simulating other quantum systems gave rise to quantum simulation. While current quantum computers are still prone to decoherence and rely on error correction, the development of hybrid algorithms that employ both quantumand classical computation allow to perform quantum simulation. Among these algorithms, the Variational Quantum Eigensolver algorithm has permitted to explore the electronic structure of simple atoms and molecules by exploiting the Rayleigh Ritz variational principle. In this work we provide the implementation of the VQE algorithm for finding the ground state energy of the hydrogen molecule on Qiskit library for python.
Feynmans ideas to employ quantum systems for simulating other quantum systems gave rise to quantum simulation. While current quantum computers are still prone to decoherence and rely on error correction, the development of hybrid algorithms that employ both quantumand classical computation allow to perform quantum simulation. Among these algorithms, the Variational Quantum Eigensolver algorithm has permitted to explore the electronic structure of simple atoms and molecules by exploiting the Rayleigh Ritz variational principle. In this work we provide the implementation of the VQE algorithm for finding the ground state energy of the hydrogen molecule on Qiskit library for python.
Abstract Solving the electronic structure for multi-electronic systems is a hard problem. Even for small atoms and molecules, approximations have to be made in order to solve numerically the Schrödinger equation. Although different methods have been developed to take into account electron correlations, their computational cost reduces their feasibility. Quantum simulation provides an alternative to traditional computational methods for enquiring the electronic structure of molecules. Specifically, the Variational Quantum Eigensolver (VQE) algorithm provides a hybrid quantum-classical algorithm for the implementation on current near term quantum devices. In this work, we explore the implementation of VQE on Qiskit for calculating the ground-state energy of diatomic Hydrogen molecule.
In this work, the underlying theory of variational quantum algorithms applied to the study of simple quantum systems has been reviewed. Using the Qiskit library, the VQE (Variational Quantum Eigensolver) algorithm has been implemented to obtain the ground state energy of three simple quantum systems: the hydrogen molecule, an electron in a Yukawa potential, and two electrons in an infinite potential well with contact interaction. The algorithm parameters that yielded the best results were the UCCSD ansatz, the COBYLA optimizer, and a high number of shots. Although the results of simulations using quantum algorithms did not achieve the same precision as classical algorithms, especially under noise models, the implementation of the VQE algorithm has been successful in exploring the current capabilities and limitations of quantum computing. These results demonstrate the ability of quantum algorithms to solve simple quantum systems, despite current challenges. Furthermore, they underscore their potential to surpass classical algorithms in solving complex quantum systems.
Quantum computing has emerged as a promising technology for solving problems that are intractable for classical computers. In this study, we introduce quantum computing and implement the Variational Quantum Eigensolver (VQE) algorithm using Qiskit on the IBM Quantum platform to calculate the ground state energy of a hydrogen molecule. We provide a theoretical framework of quantum mechanics, qubits, quantum gates, and the VQE algorithm. Our implementation process is described, and we simulate the results. Additionally, experiments are conducted on the IBM Quantum platform, and the results are analyzed. Our fi ndings demonstrate that VQE can effi ciently calculate molecular properties with high accuracy. However, limitations and challenges in scaling the algorithm for larger molecules are also identifi ed. This work contributes to the growing body of research on quantum computing and highlights the potential applications of VQE for real-world problem-solving.
Key properties of physical systems can be described by the eigenvalues of matrices that represent the system. Computational algorithms that determine the eigenvalues of these matrices exist, but they generally suffer from a loss of performance as the matrix grows in size. This process can be expanded to quantum computation to find the eigenvalues with better performance than the classical algorithms. One application of such an eigenvalue solver is to determine energy levels of a molecule given a matrix representation of its Hamiltonian using the variational principle. Using a variational quantum eigensolver, we determine the ground state energies of different molecules. We focus on the choice of optimization strategy for a Qiskit simulator on low-end hardware. The benefits of several different optimizers were weighed in terms of accuracy in comparison to an analytic classical solution as well as code efficiency.
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.
This study investigates the Variational Quantum Eigensolver (VQE), a hybrid quantum-classical algorithm that partitions the computational workload between a quantum processing unit and a classical optimiser to determine the ground-state energies of molecular systems. The Qiskit SDK v1.x framework is employed, utilising the PySCFDriver module within the qiskit-nature package to construct second-quantised molecular electronic Hamiltonians for two representative systems: Hydrogen ( ) and Lithium Hydride ( ). For the complete active space is retained without spatial reduction, mapping the full Hamiltonian onto a 4-qubit register. However for the full STO-3G representation demands 12 qubits, rendering direct classical parameter optimisation computationally intractable. This bottleneck is resolved by applying an ActiveSpaceTransformer (num_electrons=2, num_spatial_orbitals=3) to freeze the chemically inactive lithium core electrons, reducing the active register to 6 qubits. Fermionic operators are subsequently converted to Pauli strings via the Jordan-Wigner transformation ( ) enabling the quantum processor to evaluate the molecular Hamiltonian as a circuit while preserving the fermionic anti-commutation relations. Under ideal, noise-free simulation using the StatevectorEstimator primitive paired with the COBYLA optimiser (maxiter=200), all VQE energy points converge precisely onto the Full Configuration Interaction (FCI) classical baseline across the full range of interatomic separations. This confirms that the hardware-efficient TwoLocal ansatz ( rotation layers and entanglement layers with reps=3) possesses sufficient expressive power to capture the molecular ground-state wavefunction. Introducing gate-level noise through AerSimulator backends calibrated from the FakeManilaV2 and FakeNairobiV2 hardware profiles—with optimisation driven by the SPSA algorithm (maxiter=150)—reveals two critical failure modes. A systematic upward energy bias is observed, attributable to depolarising gate errors and measurement readout faults that prevent the optimiser from reaching the true minimum energy eigenvalue. Additionally, severe stochastic oscillations between 1.0 Å and 1.5 Å on the curve induce premature convergence of the SPSA optimiser to spurious local minima. These results demonstrate that unmitigated hardware noise renders VQE incapable of reliably predicting equilibrium bond lengths on current NISQ devices, motivating the future integration of Zero-Noise Extrapolation (ZNE), readout error mitigation, and Virtual Distillation into the computational pipeline. Keywords—Variational Quantum Eigensolver (VQE); Qiskit 1.x; Potential Energy Curve; Molecular Ground-State Energy; Hardware-Efficient Ansatz; Gate-Level Noise Simulation; ActiveSpaceTransformer; NISQ Devices.