Efficient diagnostics for quantum error correction

Fault-tolerant quantum computing will require accurate estimates of the resource overhead under different error correction strategies, but standard metrics such as gate fidelity and diamond distance have been shown to be poor predictors of logical performance. We present a scalable experimental approach based on Pauli error reconstruction to predict the performance of concatenated codes. Numerical evidence demonstrates that our method significantly outperforms predictions based on standard error metrics for various error models, even with limited data. We illustrate how this method assists in the selection of error correction schemes.

Figure 1

Compiling twirling (random physical Pauli) gates into fault tolerant gadgets. Panel (a) shows the noisy gates in the $k\mathrm{th}$ clock cycle of a fault tolerant quantum algorithm presented in the standard form prescribed in Ref. [11]. Twirling gates are inserted in panel (b) to tailor the noise processes to Pauli errors. These gates are compiled into existing gates by replacing easy gates by their dressed versions in panel (c).

Figure 2

Predictability of logical infidelity for level-2 concatenated Steane code. Figures 1 and 1 compare the predictive powers of our logical estimator (red) against two standard error metrics (gray): the average gate infidelity (a) and the diamond distance (b), under an ensemble of 18 000 CPTP maps. Each point $p=(x_p,y_p)$ corresponds to a physical noise process; $x_p$ is its physical error metric and $y_p$, its LER. The dispersion of points, quantified as $\mathrm{\Delta }$ in the insets, indicates the predictive power of the physical error metric. While LERs can vary over several orders of magnitude with respect to standard error metrics, our logical estimator is strongly correlated with the LER. (c) Correlated Pauli error model.

Figure 3

Accuracy of the logical estimator based on limited NR data, using a level-2 concatenated Steane code for an ensemble of about 15 000 random correlated Pauli channels. The accuracy, quantified by $\mathrm{\Delta }$, improves sharply with the number of Pauli error rates ($K$) extracted using NR. We observe that for $K=200$, which is about $1.2\%$ of all Pauli error rates on the Steane code block, the accuracy closely matches the logical estimator computed using all NR data, i.e., $K=4^7$.

Figure 4

Using the logical estimator to select an optimal code. The figure demonstrates the use of our tool in selecting an optimal error correcting code under a biased Pauli error model. The choices of codes include level-3 concatenated versions of the Steane code and the cyclic code. While the solid lines depict the values of the logical estimator, the dashed lines correspond to LERs estimated using numerical simulations. We observe that ${\tilde{p}}_u$ accurately selects the optimal code for all noise rates.

Figure 5

Randomized compiling. The top panel shows a bare circuit with alternating easy and hard cycles. The middle panel shows insertion of random Pauli gates in between easy and hard cycles. The bottom panel shows that the extra randomization gates are compiled into the existing gates resulting in a random compilation of the bare circuit [11].

Figure 6

Predicting the performance of level-2 concatenated Steane code under unitary errors. Panels (a) and (b) compare the predictive powers of our tool (red) with the standard error metrics: infidelity and the diamond distance, respectively, under an ensemble of 16 000 random unitary channels. These are similar to Figs. 2 and 2 in the main text. The dispersion in the scatter corresponding to a metric ($\mathrm{\Delta }$ in the insets) is indicative of its predictive power. The gains in predictability offered by our tool is drastic for the above case of unitary errors when compared to CPTP maps.

Figure 7

Panel (a) shows the stabilizer generators for the $3\times 3$ rotated planar code denoted by ${\mathfrak{S}}_{3\times 3}$. Panel (b) depicts an enforced concatenated structure on a rotated planar code. The blue, red, and green lattice depict levels $0,1,\text{and}2$, respectively, of the concatenated code corresponding to the underlying rotated planar code. For concatenated levels $\ell >1$, the generators in panel (a) are replaced by the corresponding logicals at level $\ell -1$.

Figure 8

Using the logical estimator to select optimal surface code. The figure demonstrates the use of our tool in selecting an optimal surface under a biased Pauli error model. The choices of codes include rotated planar code of dimensions $9\times 12$ and $16\times 9$. While the solid lines depict the values of the logical estimator, the dashed lines correspond to LERs estimated using numerical simulations. We observe that ${\tilde{p}}_u$ helps select the optimal code for all noise rates.

Figure 9

The figures highlight the rapid convergence rate of the importance sampler as compared to the direct sampler, under CPTP noise processes in Fig. 6 and coherent errors in Fig. 6. Each trend line in the figures is associated to a physical noise rate. While different colors are used to identify different physical error rates, the solid and dashed lines are used to distinguish between the sampling techniques. Note that while the direct sampler takes a large number to syndrome samples to provide a reliable estimate of $r\left({\overline{\mathcal{E}}}_{\ell }\right)$, the importance sampler achieves this task with far lesser syndrome samples. The speedup offered by importance sampling is quite drastic. The case for $r=4\times {10}^{-3}$ in Fig. 6 is a good example. The direct sampler shows signs of convergence around ${10}^7$ syndrome samples, whereas the importance sampler converges with just ${10}^4$ samples. Notice however that with only ${10}^4$ samples, the direct sampler underestimates $r\left({\overline{\mathcal{E}}}_{\ell }\right)$ by almost two orders of magnitude.