  Quantum-classical hybrid algorithms are emerging as promising candidates for
near-term practical applications of quantum information processors in a wide
variety of fields ranging from chemistry to physics and materials science. We
report on the experimental implementation of such an algorithm to solve a
quantum chemistry problem, using a digital quantum simulator based on trapped
ions. Specifically, we implement the variational quantum eigensolver algorithm
to calculate the molecular ground state energies of two simple molecules and
experimentally demonstrate and compare different encoding methods using up to
four qubits. Furthermore, we discuss the impact of measurement noise as well as
mitigation strategies and indicate the potential for adaptive implementations
focused on reaching chemical accuracy, which may serve as a cross-platform
benchmark for multi-qubit quantum simulators.
