Benchmarking Fidelity Metrics of Quantum Computers

Quantum computers are susceptible to noise or errors, and it is critical to quantify the impact of noise acting on a quantum circuit. The density matrix-based fidelity, referred to as DM Fidelity, aka Uhlmann fidelity, provides a measure that quantifies the closeness between a noise-free density matrix and its noisy density matrix. However, DM Fidelity is computationally expensive, since density matrices require full-state tomography, and also necessitates noise-free results. Consequently, different metrics have been proposed to estimate the noise impact of a quantum circuit. In this paper, we study the commonly used metrics, including Hellinger Fidelity, Total Variational Distance (TVD), Estimated Success Probability (ESP), Probability of Successful Trials (PST), Clifford-based Fidelity Estimation, purity measure using either CSWAP or Classical Shadows, and Shadow Overlap, and show how well these metrics correlate to DM Fidelity. We also identify certain regions and conditions where one metric correlates with DM Fidelity better than others. The metrics are computed with a noise simulator using noise models derived from IBMQ quantum computers on the SupermarQ benchmarks. Among the aforementioned metrics, our results show that ESP and PST have the highest correlation with DM Fidelity: ESP has a Spearman correlation of 0.88, signifying a positive nonlinear correlation, while PST has a Pearson correlation of 0.92, signifying a positive linear correlation. Accurate fidelity estimation has many applications. In our paper, we present an application in quantum error mitigation. We show that using fidelity metrics that correlate well with DM Fidelity is more effective than using other metrics. In particular, our results on real IBM Quantum hardware show that Reliability Based Zero Noise Extrapolation (RZNE), using ESP/PST achieves a fidelity (Hellinger) improvement of 7.44x/7.83x over the baseline without error-mitigation. Using other metrics like Clifford fidelity which has a Spearman Correlation of 0.73 to DM Fidelity, would be less effective: RZNE with Clifford fidelity shows a fidelity improvement gain of 4.31x.