  Non-Markovian effects are important in modeling the behavior of open quantum
systems arising in solid-state physics, quantum optics as well as in study of
biological and chemical systems. The non-Markovian environment is often
approximated by discrete bosonic modes, thus mapping it to a Lindbladian or
Hamiltonian simulation problem. While systematic constructions of such modes
have been previously proposed, the resulting approximation lacks rigorous and
general convergence guarantees. In this letter, we show that under some
physically motivated assumptions on the system-environment interaction, the
finite-time dynamics of the non-Markovian open quantum system computed with a
sufficiently large number of modes is guaranteed to converge to the true
result. Furthermore, we show that this approximation error typically falls off
polynomially with the number of modes. Our results lend rigor to classical and
quantum algorithms for approximating non-Markovian dynamics.
