  Quantum collision models are receiving increasing attention as they describe
many nontrivial phenomena in dynamics of open quantum systems. In a general
scenario of both fundamental and practical interest, a quantum system
repeatedly interacts with individual particles or modes forming a correlated
and structured reservoir; however, classical and quantum environment
correlations greatly complicate the calculation and interpretation of the
system dynamics. Here we propose an exact solution to this problem based on the
tensor network formalism. We find a natural Markovian embedding for the system
dynamics, where the role of an auxiliary system is played by virtual indices of
the network. The constructed embedding is amenable to analytical treatment for
a number of timely problems like the system interaction with two-photon
wavepackets, structured photonic states, and one-dimensional spin chains. We
also derive a time-convolution master equation and relate its memory kernel
with the environment correlation function, thus revealing a clear physical
picture of memory effects in the dynamics. The results advance tensor-network
methods in the fields of quantum optics and quantum transport.
