Using variational quantum eigensolver (VQE) algorithm to realize ground state energy calculation on noisy intermediate-scale quantum (NISQ) devices is widely applied in quantum physics, quantum chemistry, and other fields. It can simulate chemical molecules to solve the energy solution required by chemical reactions. This paper proposes an optimized cost function for the original meta-VQE and combines it with the quantum architecture search algorithm to break the limitation of the fixed structure. To expand the penalty degree of error and obtain more accurate results, two indicators are added to the original cost function including the slope error of discrete predicted value and accurate value as well as the error of the first value. The results show that the optimized algorithm can learn more characteristics of the ground state energy changing trend in the noise channel, and obtain a more accurate estimate of equilibrium bond length.
Using variational quantum eigensolver (VQE) algorithm to realize ground state energy calculation on noisy intermediate-scale quantum (NISQ) devices is widely applied in quantum physics, quantum chemistry, and other fields. It can simulate chemical molecules to solve the energy solution required by chemical reactions. This paper proposes an optimized cost function for the original meta-VQE and combines it with the quantum architecture search algorithm to break the limitation of the fixed structure. To expand the penalty degree of error and obtain more accurate results, two indicators are added to the original cost function including the slope error of discrete predicted value and accurate value as well as the error of the first value. The results show that the optimized algorithm can learn more characteristics of the ground state energy changing trend in the noise channel, and obtain a more accurate estimate of equilibrium bond length.
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.
Abstract The variational quantum eigensolver (VQE) is a hybrid quantum–classical algorithm for preparing ground states in the current era of noisy devices. The classical component of the algorithm requires a large number of measurements on intermediate parameter values that are typically discarded. However, intermediate steps across many calculations can contain valuable information about the relationship between the quantum circuit parameters, resultant measurements, and noise specific to the device. This information would be useful when analysing families of Hamiltonians such as for potential energy landscapes for molecular interaction with catalytic surfaces or confinement in metal–organic-frameworks. In this work, we use supervised machine learning on the intermediate parameter and measurement data to amortise prediction of optimal final parameters across families of Hamiltonians. Our technique optimises parameters leading to chemically accurate ground state energies much faster than conventional techniques. It requires significantly fewer iterations and simultaneously shows the ability to find noise-aware angles if trained on noisy devices even with noise drift. We demonstrate this technique on IBM quantum devices by predicting ground state energies of H 2 for one and two qubits; H 3 for three qubits; and HeH + for four qubits where it finds optimal angles using only modelled data for training. Our technique amortises classical optimisation across multiple VQE runs, resulting in significantly faster overall classical optimisation without additional quantum overhead compared to running VQE.
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.
Quantum chemistry is one of the most promising applications of quantum computers in the near future. For noisy intermediate-scale quantum devices, the quantum-classical hybrid framework based on the variational quantum eigensolver (VQE) has become the method of choice. In the literature, there are many different variants of VQE, but it is not known which one is optimal for a given molecule. For this purpose, we perform a thorough benchmarking on more than ten different kinds of VQE ansatzes (in systems up to 30 qubits), based on their performance on the energy accuracy, runtime until convergence, and number of parameters. Our results show that the ADAPT ansatz can be used to obtain more accurate energy for small systems (below 14 qubits), but it costs much more computational resources. For larger molecules, UCCSD0 has better performance. However, all the tested ansatzes can hardly reach chemical accuracy at stretched bond lengths. Our results were obtained using MindSpore Quantum, where the codes and the benchmarking toolkit are publicly available at Gitee.
Variational quantum eigensolver (VQE) is a hybrid quantum-classical algorithm designed for noisy intermediate-scale quantum (NISQ) computers. It is promising for quantum chemical calculations (QCC) because it can calculate the ground-state energy of a target molecule. Although VQE has a potential to achieve a higher accuracy than classical approximation methods in QCC, it is challenging to achieve it on current NISQ computers due to the significant impact of noises. Density matrix embedding theory (DMET) is a well-known technique to divide a molecule into multiple fragments, which is available to mitigate the noise impact on VQE. However, our preliminary evaluation shows that the naive combination of DMET and VQE does not outperform a gold standard classical method. In this work, we present three approaches to mitigate the noise impact for the DMET+VQE combination. (1) The size of quantum circuits used by VQE is decreased by reducing the number of bath orbitals which represent interactions between multiple fragments in DMET. (2) Reduced density matrices (RDMs), which are used to calculate a molecular energy in DMET, are calculated accurately based on expectation values obtained by executing quantum circuits using a noise-less quantum computer simulator. (3) The parameters of a quantum circuit optimized by VQE are refined with mathematical post-processing. The evaluation using a noisy quantum computer simulator shows that our approaches significantly improve the accuracy of the DMET+VQE combination. Moreover, we demonstrate that on a real NISQ device, the DMET+VQE combination applying our three approaches achieves a higher accuracy than the gold standard classical method.
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.