  We study the dynamics of chiral quantum networks consisting of nodes coupled
by unidirectional or asymmetric bidirectional quantum channels. In contrast to
familiar photonic networks where driven two-level atoms exchange photons via 1D
photonic nanostructures, we propose and study a setup where interactions
between the atoms are mediated by spin excitations (magnons) in 1D $XX$ spin
chains representing spin waveguides. While Markovian quantum network theory
eliminates quantum channels as structureless reservoirs in a Born-Markov
approximation to obtain a master equation for the nodes, we are interested in
non-Markovian dynamics. This arises from the nonlinear character of the
dispersion with band-edge effects, and from finite spin propagation velocities
leading to time delays in interactions. To account for the non-Markovian
dynamics we treat the quantum degrees of freedom of the nodes and connecting
channels as a composite spin system with the surrounding of the quantum network
as a Markovian bath, allowing for an efficient solution with time-dependent
density matrix renormalization group techniques. We illustrate our approach
showing non-Markovian effects in the driven-dissipative formation of quantum
dimers, and we present examples for quantum information protocols involving
quantum state transfer with engineered elements as basic building blocks of
quantum spintronic circuits.
