Simplest fidelity-estimation method for graph states with depolarizing noise

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.