  One of the main frameworks to analyze the effects of the environment in a
quantum computer is that of pure dephasing, where the dynamics of qubits can be
characterised in terms of a well-known dynamical map. In this work we present a
non-peturbative extension of such map beyond this simple pure-dephasing case,
i.e. that is valid for a general spin coupled to a bosonic environment in a
thermal state. To this aim, we use a Trotter decomposition and a Magnus
expansion to simplify the unitary evolution operator in interaction picture.
The proposed derivation can be extended to other finite-level open quantum
systems including many body, initial system-environment correlated states,
multiple-time correlation functions or quantum information protocols.
