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 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.