  Enabled by rapidly developing quantum technologies, it is possible to network
quantum systems at a much larger scale in the near future. To deal with
non-Markovian dynamics that is prevalent in solid-state devices, we propose a
general transfer function based framework for modeling linear quantum networks,
in which signal flow graphs are applied to characterize the network topology by
flow of quantum signals. We define a noncommutative ring $\mathbb{D}$ and use
its elements to construct Hamiltonians, transformations and transfer functions
for both active and passive systems. The signal flow graph obtained for direct
and indirect coherent quantum feedback systems clearly show the feedback loop
via bidirectional signal flows. Importantly, the transfer function from input
to output field is derived for non-Markovian quantum systems with colored
inputs, from which the Markovian input-output relation can be easily obtained
as a limiting case. Moreover, the transfer function possesses a symmetry
structure that is analogous to the well-know scattering transformation in \sd
picture. Finally, we show that these transfer functions can be integrated to
build complex feedback networks via interconnections, serial products and
feedback, which may include either direct or indirect coherent feedback loops,
and transfer functions between quantum signal nodes can be calculated by the
Riegle's matrix gain rule. The theory paves the way for modeling, analyzing and
synthesizing non-Markovian linear quantum feedback networks in the
frequency-domain.
