Learning quantum graph states with product measurements

We consider the problem of learning N identical copies of an unknown n-qubit quantum graph state with product measurements. These graph states have corresponding graphs where every vertex has exactly d neighboring vertices. Here, we detail an explicit algorithm that uses product measurements on multiple identical copies of such graph states to learn them. When n ≫ d and N = O(d log(1/ϵ) + d² log n), this algorithm correctly learns the graph state with probability at least 1 – ϵ. From channel coding theory, we find that for arbitrary joint measurements on graph states, any learning algorithm achieving this accuracy requires at least Ω(log(1/ϵ) + d log n) copies when $d = o\left( {\sqrt n } \right)$. We also supply bounds on N when every graph state encounters identical and independent depolarizing errors on each qubit.