  Graph states are entangled states useful for several quantum information
processing tasks such as measurement-based quantum computation and quantum
metrology. As the size of graph states realized in experiments increases, it
becomes more essential to devise efficient methods estimating the fidelity
between the ideal graph state and an experimentally-realized actual state. Any
efficient fidelity-estimation method, in general, must use multiple
experimental settings, i.e., needs to switch between at least two measurements.
Recently, it has been shown that a single measurement is sufficient if the
noise can be modeled as the phase-flip error. Since the bit-flip error should
also occur in several experiments, it is desired to extend this simplest method
to noise models that include phase and bit-flip errors. However, it seems to be
nontrivial because their result strongly depends on properties of the
phase-flip error. In this paper, by analyzing effects of the bit-flip error on
stabilizer operators of graph states, we achieve the extension to the
depolarizing noise, which is a major noise model including phase and bit-flip
errors. We also numerically evaluate our simplest method for noise models
interpolating between the phase-flip and depolarizing noises.
