  We propose a numerical technique based on a combination of short-iterative
Lanczos and exact diagonalization methods, suitable for simulating the time
evolution of the reduced density matrix of a single qubit interacting with an
environment. By choosing a mode discretization method and a flexible bath
states truncation scheme, we are able to include in the physical description
multiple-excitation processes, beyond weak coupling and Markov approximations.
We apply our technique to the simulation of three different model Hamiltonians,
which are relevant in the field of adiabatic quantum computation. We compare
our results with those obtained on the basis of the widely used Lindblad master
equation, as well as with well-known exact and approximated approaches. We show
that our method is able to recover the thermodynamic behavior of the qubit-bath
system, beyond the Born-Markov approximation. Finally, we show that even in the
case of the adiabatic quantum annealing of a single qubit the bath can be
beneficial in reaching the reduced system ground state.
