Directly estimating the fidelity of measurement-based quantum computation

In measurement-based quantum computation (MBQC), quantum circuits are implemented using adaptive measurements on an entangled resource state. In practice, the resource state will always be prepared with some noise, and it is crucial to understand the effect of this noise on the operation of MBQC. Typically, one measures the fidelity of the noisy resource state with the ideal resource state with the assumption that a high fidelity state means a high fidelity computation. However, the precise relationship between these two fidelities is not known. Here, we derive an expression that equates the average fidelity of MBQC to a certain correlation function evaluated on the noisy resource state. Using this expression, we show that state fidelity provides a tight lower bound on average MBQC fidelity. Conversely, we find that state fidelity can also greatly underestimate average MBQC fidelity, even for realistic noise channels, implying that state fidelity is not a good indicator of MBQC performance in general. To address this, we formulate an efficient method to directly estimate average MBQC fidelity by measuring the aforementioned correlation function. These results therefore improve our ability to characterize noisy resource states in quantum computers and benchmark MBQC performance.