We present numerical calculations, and simulations performed on a Rydberg atom quantum simulator, of the adiabatic evolution of closed many-body quantum systems around a quantum phase transition. We demonstrate that the end-to-end transfer error, for a given process duration, can be suppressed by adopting smooth initial and final scheduling functions for the Hamiltonian. We consider a one-dimensional mixed-field Ising model, as well as a chain of Rydberg atoms, and compare numerical calculations and results on a commercially available quantum computer for the end-to-end transfer error with different schedule functions. We provide a new rigorous proof under minimal regularity assumptions of the known result that if the time dependent Hamiltonian is $n$ times differentiable with vanishing first to $n$ th order derivatives in the beginning and in the end, the infidelity with respect to the final adiabatic eigenstate scales as $1/T^{n+1}$ when evolving for time $T$.

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• Adiabatic quantum optimization
• Rydberg atoms & molecules
• Adiabatic approximation
• Density matrix renormalization group
• Ising model
• Tensor network methods