  We derive an exact and explicit Kraus decomposition for the reduced density
of a quantum system simultaneously interacting with time-dependent external
fields and a chaotic environment of thermodynamic dimension. We test the
accuracy of the Kraus decomposition against exact numerical results for a CNOT
gate performed on two qubits of an $(N+2)$-qubit statically flawed isolated
quantum computer. Here the $N$ idle qubits comprise the finite environment. We
obtain very good agreement even for small $N$.
