Quantum Private Information Retrieval on Forests

We consider the problem of graph private information retrieval (PIR) in the quantum setting. We focus on a specific type of graphs, namely the forest graphs. A forest graph is a union of disjoint trees. We consider the quantum setting where all the servers share an entangled state. Each server is allowed to send only one qudit per retrieval round. We design a scheme for the forest graphs that achieves the rate of $\frac{3}{N}$. The scheme uses a GHZ state as the initial entangled state between the servers prior to the scheme initiation. We compare our scheme with the only work done in the literature that considered the quantum setting of graph PIR. We show that our scheme is more efficient in this setting. We derive the quantum capacity for the case of $G=3K_2$, which is equal to $4/6$. $G=3K_2$ is a special case of the forest graph. We design a scheme that is specialized for this graph, achieving a rate equal to $4/6$ and provide the corresponding matching upper bound, establishing it as the capacity.