  In this paper we propose a technique for distributing entanglement in
architectures in which interactions between pairs of qubits are constrained to
a fixed network $G$. This allows for two-qubit operations to be performed
between qubits which are remote from each other in $G$, through gate
teleportation. We demonstrate how adapting \emph{quantum linear network coding}
to this problem of entanglement distribution in a network of qubits can be used
to solve the problem of distributing Bell states and GHZ states in parallel,
when bottlenecks in $G$ would otherwise force such entangled states to be
distributed sequentially. In particular, we show that by reduction to classical
network coding protocols for the $k$-pairs problem or multiple multicast
problem in a fixed network $G$, one can distribute entanglement between the
transmitters and receivers with a Clifford circuit whose quantum depth is some
(typically small and easily computed) constant, which does not depend on the
size of $G$, however remote the transmitters and receivers are, or the number
of transmitters and receivers. These results also generalise straightforwardly
to qudits of any prime dimension. We demonstrate our results using a
specialised formalism, distinct from and more efficient than the stabiliser
formalism, which is likely to be helpful to reason about and prototype such
quantum linear network coding circuits.
