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 $n \gg d$ and $N = O(d \log(1/\epsilon) + d^2 \log n ),$ this algorithm correctly learns the graph state with probability at least $1- \epsilon$. From channel coding theory, we find that for arbitrary joint measurements on graph states, any learning algorithm achieving this accuracy requires at least $\Omega(\log (1/\epsilon) + d \log n)$ copies when $d=o(\sqrt n)$. We also supply bounds on $N$ when every graph state encounters identical and independent depolarizing errors on each qubit.