Quantum linear network coding for entanglement distribution in restricted architectures

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.