Effect of noise correlations on randomized benchmarking

Among the most popular and well-studied quantum characterization, verification, and validation techniques is randomized benchmarking (RB), an important statistical tool used to characterize the performance of physical logic operations useful in quantum information processing. In this work we provide a detailed mathematical treatment of the effect of temporal noise correlations on the outcomes of RB protocols. We provide a fully analytic framework capturing the accumulation of error in RB expressed in terms of a three-dimensional random walk in “Pauli space.” Using this framework we derive the probability density function describing RB outcomes (averaged over noise) for both Markovian and correlated errors, which we show is generally described by a $\mathrm{\Gamma }$ distribution with shape and scale parameters depending on the correlation structure. Long temporal correlations impart large nonvanishing variance and skew in the distribution towards high-fidelity outcomes—consistent with existing experimental data—highlighting potential finite-sampling pitfalls and the divergence of the mean RB outcome from worst-case errors in the presence of noise correlations. We use the filter-transfer function formalism to reveal the underlying reason for these differences in terms of effective coherent averaging of correlated errors in certain random sequences. We conclude by commenting on the impact of these calculations on the utility of single-metric approaches to quantum characterization, verification, and validation.

Figure 1

Analytic PDFs $〈\mathcal{F}〉$ and simulated distributions for different noise regimes using $p_{\text{RB}}\sim 2\times {10}^{-4}$. (a),(b) $f_{〈\mathcal{F}〉}\left(F\right)$ calculated as a function of $J$ for Markovian and DC regimes, normalized to unity at the mode for each $J$ for clarity. The white line indicates analytic mean, $\mathbb{E}\left[〈\mathcal{F}〉\right]$; the red dashed line indicates analytic mode, $\mathbb{M}\left[〈\mathcal{F}〉\right]$. (c),(d) Comparison of analytics (solid lines) to numerically simulated histograms of distributions for various $J$, using $n=50$ and no free parameters. Curves are vertically offset by for clarity by multiples of 25 units. Inset in (c): Markovian distributions varying $n=10$, 50, 250 (black, blue, pink), corresponding to distribution in dotted box. (e) Analytic PDF $〈\mathcal{F}〉$ (solid red line) and simulated distributions for quasi-$1/\omega$ noise regime with $J=200$ and $n=3000$. Comparative PDF for DC (left axis) and Markovian noise (right axis) are shown as black dashed and solid lines, respectively. Noise parameters are chosen such that the mean error is $0.05$. PSD shown in inset is constructed using Fourier synthesis as described in [29].

Figure 2

Filter functions $\left\{G\right(\omega ;\mathbf{\eta }\left)\right\}$ for ensemble of RB sequences. Dimensionless angular frequency $\omega$ normalized to the duration of a bit-flip operation, ${\tau }_{\pi }$; low-frequency ($\omega <1$) noise susceptibility is captured by vertical offsets and slopes. (Inset) Histogram of calculated fidelities for $\left\{G\right(\omega ;\mathbf{\eta }\left)\right\}$ using $S\left(\omega \right)\propto \delta (\omega -4\pi \times {10}^{-4}{\tau }_{\pi }^{-1})$ for quasi-DC noise (scaled to correspond to $\sigma =0.015$) and overlaid with $f_{〈\mathcal{F}〉}\left(F\right)$.