molecule, emphasizing practical implementation and performance on current quantum hardware. Our research presents a comparative study of HEA and UCCSD ansätze on noiseless and noisy simulations and implements VQE on recent IBM quantum computer noise models and a real quantum computer, IBM Fez, providing a fully functional code employing Qiskit 1.2. Our experiments confirm UCCSD's reliability in ideal conditions, while the HEA demonstrates greater robustness to hardware noise, achieving chemical accuracy on state-vector simulation (SVS). The results reveal that achieving ground-state energy within chemical accuracy is feasible without error mitigation during VQE convergence. We demonstrate that current quantum devices effectively optimize circuit parameters despite misestimating simulated energies. The SVS-evaluated energies provide a more accurate representation of the solution quality compared to QPU-estimated energy values, indicating that VQE converges to the correct ground state despite quantum noise. Our study also applies noise mitigation as a postprocessing technique, using zero-noise extrapolation (ZNE) on a real quantum computer. The detailed methodologies presented in this study, including Hamiltonian construction and Fermionic-to-qubit transformations, facilitate flexible adaptation of the VQE approach for various algorithm variants and across different levels of algorithmic implementation.
molecule, emphasizing practical implementation and performance on current quantum hardware. Our research presents a comparative study of HEA and UCCSD ansätze on noiseless and noisy simulations and implements VQE on recent IBM quantum computer noise models and a real quantum computer, IBM Fez, providing a fully functional code employing Qiskit 1.2. Our experiments confirm UCCSD's reliability in ideal conditions, while the HEA demonstrates greater robustness to hardware noise, achieving chemical accuracy on state-vector simulation (SVS). The results reveal that achieving ground-state energy within chemical accuracy is feasible without error mitigation during VQE convergence. We demonstrate that current quantum devices effectively optimize circuit parameters despite misestimating simulated energies. The SVS-evaluated energies provide a more accurate representation of the solution quality compared to QPU-estimated energy values, indicating that VQE converges to the correct ground state despite quantum noise. Our study also applies noise mitigation as a postprocessing technique, using zero-noise extrapolation (ZNE) on a real quantum computer. The detailed methodologies presented in this study, including Hamiltonian construction and Fermionic-to-qubit transformations, facilitate flexible adaptation of the VQE approach for various algorithm variants and across different levels of algorithmic implementation.
While numerical simulations are presented in most papers introducing new methods to enhance the VQE performance, comprehensive, comparative, and applied studies remain relatively rare. We present a comprehensive, yet concise guide for the implementation of the VQE for molecular problems on NISQ devices, specifically applied to estimate the ground state energy of the BeH2 molecule using hardware-efficient and chemically informed ansätze. This work clarifies several under-documented aspects in the literature, such as the construction of the electronic Hamiltonian, the transformation of fermionic operators into qubit operators via second quantization, and the mathematical framework's details for the unitary coupled cluster single and double (UCCSD) ansatz. Our methodology, implemented using Qiskit 1.2, the latest release as of the date of this writing, is demonstrated on a noiseless simulator and further tested with noisy quantum circuits. The resilience of the VQE to quantum noise remains an open question. This study compares the computational accuracy of ground state energy estimations for molecules using the VQE across three different current quantum hardware noise models. Furthermore, our experiment on IBM's 156-qubit actual quantum computer revealed valuable insights on the real performance of the VQE on current quantum hardware.
Variational quantum eigensolvers (VQEs) are among the most promising quantum algorithms for solving electronic structure problems in quantum chemistry, particularly during the noisy intermediate-scale quantum (NISQ) era. In this study, we investigate the capabilities and limitations of VQE algorithms implemented on current quantum hardware for determining molecular ground-state energies, focusing on the adaptive derivative-assembled pseudo-Trotter ansatz VQE (ADAPT-VQE). To address the significant computational challenges posed by molecular Hamiltonians, we explore various well known strategies to simplify the Hamiltonian, optimize the ansatz, and improve classical parameter optimization through modifications of the COBYLA optimizer. These enhancements are integrated into a tailored quantum computing implementation designed to minimize the circuit depth and computational cost. Using benzene as a benchmark system, we demonstrate the application of these optimizations on an IBM quantum computer. Despite these improvements, our results highlight the limitations imposed by current quantum hardware, particularly the impact of quantum noise on state preparation and energy measurement. The noise levels in today's devices prevent meaningful evaluations of molecular Hamiltonians with sufficient accuracy to produce reliable quantum chemical insights. Finally, we extrapolate the requirements for future quantum hardware to enable practical and scalable quantum chemistry calculations using VQE algorithms. This work provides an assessment of current quantum algorithms for molecular modeling on real quantum hardware, highlighting the impact of noise and hardware limitations on the achievable accuracy.
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.
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.
Abstract Quantum computers are expected to be highly beneficial for chemistry simulations, promising significant improvements in accuracy and speed. The most prominent algorithm for chemistry simulations on NISQ devices is the Variational Quantum Eigensolver (VQE). It is a hybrid quantum-classical algorithm which calculates the ground state energy of a Hamiltonian based on parametrized quantum circuits, while a classical optimizer is used to find optimal parameter values. However, quantum hardware is affected by noise, and it needs to be understood to which extent it can degrade the performance of the VQE algorithm. In this paper, we study the impact of noise on the example of the hydrogen molecule. First, we compare the VQE performance for a set of various optimizers, from which we find NFT to be the most suitable one. Next, we quantify the effect of different noise sources by systematically increasing their strength. The noise intensity is varied around values common to superconducting devices of IBM Q, and curve fitting is used to model the relationship between the obtained energy values and the noise magnitude. Since the amount of noise in a circuit highly depends on its architecture, we perform our studies for different ansatzes, including both hardware-efficient and chemistry-inspired ones. Finally, we compare our results from noisy simulations with experiments on different IBM hardware devices.
Abstract Quantum computers promise to solve electronic structure, lattice dynamics, and strongly correlated problems beyond classical reach. However, near-term noisy intermediate scale quantum (NISQ) processors require systematic benchmarking to establish where variational algorithms with error mitigation can achieve chemical accuracy and cor rectly predict physical phenomena. Here we present 28 quantum material simula tions implemented with Qiskit, spanning molecular hydrogen (H 2 ) through the iron molybdenum cofactor (FeMoco), one-dimensional Hubbard chains, Kitaev honeycomb spin liquids, topological insulators (SSH model), BCS superconductors, and disordered Anderson insulators. Using the variational quantum eigensolver (VQE) with UCCSD ansatz, we achieve chemical accuracy (±1.6 mHa) for H 2 across the full dissociation curve and recover exact ground-state energies for H4 within 9 mHa after zero-noise extrapolation (ZNE) — reducing raw VQE errors from 152 mHa to 9 mHa. We accu rately identify the transverse-field Ising quantum critical point (h c = J) with 4 qubits, resolve Majorana Dirac dispersions in the Kitaev spin liquid, and reproduce the Mott transition in the Fermi-Hubbard model at U c = 6t. Resource estimation for FeMoco reveals O(N 2.5 ) scaling, projecting chemical accuracy with ∼200 logical qubits. Across all benchmarks, we quantify the accuracy-circuit depth tradeoff, demonstrating that depth-6 VQE ans¨atze achieve chemical accuracy for small molecules while depth-10 is required for correlated lattices. These results establish a comprehensive benchmark suite for evaluating quantum hardware, error mitigation protocols, and variational al gorithms for materials science on NISQ processors.