  Verifying quantum states is central to certifying the correct operation of
various quantum information processing tasks. In particular, in
measurement-based quantum computing, checking whether correct graph states are
generated is essential for reliable quantum computing. Several verification
protocols for graph states have been proposed, but none of these are
particularly resource efficient: multiple copies are required to extract a
single state that is guaranteed to be close to the ideal one. The best protocol
currently known requires $O(n^{15})$ copies of the state, where $n$ is the size
of the graph state. In this paper, we construct a significantly more
resource-efficient verification protocol for graph states that only requires
$O(n^5\log{n})$ copies. The key idea is to employ Serfling's bound, which is a
probability inequality in classical statistics. Utilizing Serfling's bound also
enables us to generalize our protocol for qudit and continuous-variable graph
states. Constructing a resource-efficient verification protocol for them is
non-trivial. For example, the previous verification protocols for qubit graph
states that use the quantum de Finetti theorem cannot be generalized to qudit
and continuous-variable graph states without tremendously increasing the
resource overhead. This is because the overhead caused by the quantum de
Finetti theorem depends on the local dimension. On the other hand, in our
protocol, the resource overhead is independent of the local dimension, and
therefore generalizing to qudit or continuous-variable graph states does not
increase the overhead. The flexibility of Serfling's bound also makes our
protocol robust: our protocol accepts slightly noisy but still useful graph
states.
