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Best-practice aspects of quantum-computer calculations: A case study of hydrogen molecule

Quantum computers are reaching one crucial milestone after another. Motivated by their progress in quantum chemistry, we have performed an extensive series of simulations of quantum-computer runs that were aimed at inspecting best-practice aspects of these calculations. In order to compare the performance of different set-ups, the ground-state energy of hydrogen molecule has been chosen as a benchmark for which the exact solution exists in literature. Applying variational quantum eigensolver (VQE) to a qubit Hamiltonian obtained by the Bravyi-Kitaev transformation we have analyzed the impact of various computational technicalities. These include (i) the choice of optimization methods, (ii) the architecture of quantum circuits, as well as (iii) 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 SPSA and COBYLA optimization methods have clearly outperformed the Nelder-Mead and Powell methods. The results obtained when using the $R_{\mathrm{y}}$ variational form were better than those obtained when the $R_{\mathrm{y}}R_{\mathrm{z}}$ form was used. The choice of an optimum {entangling layer} was sensitively interlinked with the choice of the optimization method. The circular {entangling layer} has been found to worsen the performance of the COBYLA method while the full {entangling layer} improved it. All four optimization methods sometimes lead to an energy that corresponds to an excited state rather than the ground state. We also show that a similarity analysis of measured probabilities can provide a useful insight.