  Graph states are entangled resource states for universal measurement-based
quantum computation. Although matter qubits such as superconducting circuits
and trapped ions are promising candidates to generate graph states, it is
technologically hard to entangle a large number of them due to several types of
noise. Since they must be sufficiently cooled to maintain their quantum
properties, thermal noise is one of major ones. In this paper, we show that for
any temperature $T$, the fidelity $\langle G|\rho_T|G\rangle$ between an ideal
graph state $|G\rangle$ at zero temperature and a thermal graph state $\rho_T$,
which is a graph state at temperature $T$, can be efficiently estimated by
using only one measurement setting. A remarkable property of our protocol is
that it is passive, while existing protocols are active, namely they switch
between at least two measurement settings. Since thermal noise is equivalent to
an independent phase-flip error, our estimation protocol also works for that
error. By generalizing our protocol to hypergraph states, we apply our protocol
to the quantum-computational-supremacy demonstration with instantaneous quantum
polynomial time circuits. Our results should make the characterization of
entangled matter qubits extremely feasible under thermal noise.
