Best-Practice Aspects of Quantum-Computer Calculations: A Case Study of the Hydrogen Molecule

Abstract

Quantum computers are reaching one crucial milestone after another. Motivated by their progress in quantum chemistry, we performed an extensive series of simulations of quantum-computer runs that were aimed at inspecting the best-practice aspects of these calculations. In order to compare the performance of different setups, the ground-state energy of the hydrogen molecule was chosen as a benchmark for which the exact solution exists in the literature. Applying the variational quantum eigensolver (VQE) to a qubit Hamiltonian obtained by the Bravyi–Kitaev transformation, we analyzed the impact of various computational technicalities. These included (i) the choice of the optimization methods, (ii) the architecture of the quantum circuits, as well as (iii) the different types of noise when simulating real quantum processors. On these, we eventually performed a series of experimental runs as a complement to our simulations. The simultaneous perturbation stochastic approximation (SPSA) and constrained optimization by linear approximation (COBYLA) optimization methods clearly outperformed the Nelder–Mead and Powell methods. The results obtained when using the Ry variational form were better than those obtained when the RyRz form was used. The choice of an optimum entangling layer was sensitively interlinked with the choice of the optimization method. The circular entangling layer was found to worsen the performance of the COBYLA method, while the full-entangling layer improved it. All four optimization methods sometimes led to an energy that corresponded to an excited state rather than the ground state. We also show that a similarity analysis of measured probabilities can provide a useful insight.

Evaluating the impact of noise on the performance of the Variational Quantum Eigensolver

“…This discrete behavior has also been observed in other works, e.g. in Miháliková et al (2022a), and may indicate excited energy levels of the Hamiltonian. Compared to NFT, the modified SPSA shows more fluctuations in its results around these two levels due to its stochastic nature.…”

“…The suitability of different classical optimizers for noisy VQE simulations was investigated in the works by Miháliková et al (2022a); Gowrishankar et al (2022). In Miháliková et al (2022a) the accuracy of H 2 simulations was compared for the four optimizers Simultaneous Perturbation Stochastic Approximation (SPSA), COBYLA, Nelder-Mead and Powell. The most accurate results were achieved with SPSA and COBYLA.…”

“…Except for Miháliková et al (2022a), the presented optimizers were tested for other problems than the ground state energy calculation of hydrogen. In our studies, we thus build up upon Miháliková et al (2022a) and investigate the performance for hydrogen using additional optimizers, such as NFT and the Bayesian optimizer.…”

“…This includes the optimizers Nelder-Mead Nelder and Mead (1965); Virtanen et al (2020) and SPSA Spall, J. C. (1998); Virtanen et al (2020), which are frequently used for VQE problems, e.g. in the works by Peruzzo et al (2014); Kandala et al (2017); Miháliková et al (2022a). Based on the discussions in section 2, we further take into account NFT Nakanishi et al (2020) and the Bayesian optimizer Kushner (1963); Head et al (2018), as these are expected to be highly resilient to noise.…”

Applications of noisy quantum computing and quantum error mitigation to “adamantaneland”: a benchmarking study for quantum chemistry

“…[53][54][55] Most VQE applications in chemistry, including proof-ofprinciple demonstrations or benchmarking studies, have focussed on determining absolute energies and potential energy curves resulting from bond dissociation or spatial deformations. [56][57][58] The research has also predominantly concentrated on analyzing small molecules such as hydrogen, 59,60 halides, 61 hydrides, 62 water, 63 and others. [64][65][66][67] Rather than absolute quantities, the most interesting chemical applications involve comparing properties derived from energy differences between two or more electronic states or geometries and molecules larger than a few atoms.…”