Improved quantum error correction with randomized compiling

Current hardware for quantum computing suffers from high levels of noise, and so to achieve practical fault-tolerant quantum computing will require powerful and efficient methods to correct for errors in quantum circuits. Here, we explore the role and effectiveness of using noise tailoring techniques to improve the performance of error correcting codes. Noise tailoring methods such as randomized compiling (RC) convert complex coherent noise processes to effective stochastic noise. While it is known that this can be leveraged to design efficient diagnostic tools, we explore its impact on the performance of error correcting codes. Of particular interest is the important class of coherent errors, arising from control errors, where RC has the maximum effect—converting these into purely stochastic errors. For these errors, we show here that RC delivers an improvement in performance of the concatenated Steane code by several orders of magnitude. We also show that below a threshold rotation angle, the gains in logical fidelity can be arbitrarily magnified by increasing the size of the codes. These results suggest that using randomized compiling can lead to a significant reduction in the resource overhead required to achieve fault tolerance.

Figure 1

The above figure shows that the gain at level $\ell$ and the gain ${\delta }_{\ell }$ scales doubly exponentially with $\ell$. The rotation angle used here is $\omega =\pi /20$.

Figure 2

Gains in logical performance ${\delta }_{\ell }$ of a level $\ell$ concatenated Steane code for rotations by angle $\omega$ about the $Z$ axis. The common crossover point lies at ${\omega }_{★}\approx 0.51$, which corresponds to a rotation angle of about ${15}^{\circ }$, below which gains from RC can be amplified by increasing the number of levels of concatenation.

Figure 3

The average gain in performance from RC, using the Haar average over all axes of rotation, for the level $\ell$ concatenated Steane code. The average gains are larger for small magnitudes of rotation. We observe that the gains increase significantly with the number of levels for $\omega \le {\overline{\omega }}_{★}\approx 0.65$, which corresponds to a rotation angle of about ${19}^{\circ }$.

Figure 4

The threshold angle ${\omega }_{★}$ for which ${\delta }_2<{\delta }_1$ for rotations about an axis parameterized by $\theta ,\varphi$. Each point on the sphere has coordinates $\{sin(\theta )cos(\varphi ),sin(\theta )sin(\varphi ),cos(\theta \left)\right\}$ and the color denotes the threshold value of angle $\omega$ for which the above condition holds. It shows that the cardinal axes do not have the highest threshold.

Figure 5

The impact of the depolarizing component on the gains from RC. We fix the average infidelity per qubit to be $r\approx 0.003$ and increase the value of the depolarizing strength from $p={10}^{-4}$ to $p={10}^{-3}$. The value of $\omega$ corresponding to each value of $p$ is chosen such that the total physical infidelity of the qubit remains constant. We observe that the gains from RC diminish with increase in depolarizing strength. This is because RC does not impact the stochastic component of the noise model.

Figure 6

The above figures highlight the strong dependence of the impact of RC on the details of the physical noise process, for concatenated Steane codes. The ensemble of noise processes considered here comprises of 16000 samples of unitary rotations about a fixed random axis. Red and green points are used to identify physical noise processes that lead to a performance gain and a performance loss, respectively, in the presence of RC. The magnitude of performance gains and losses are measured by the ratio of logical error rates in the non-RC and RC settings, i.e., ${\delta }_1$ for level-1 concatenated Steane code in Fig. 6, and ${\delta }_2$ for level-2 in Fig. 6.

Figure 7

The above figures highlight the strong dependence of the impact of RC on the details of the physical noise process, for concatenated Steane codes. The ensemble of noise processes considered here comprises of $18000$ random CPTP maps. Red and green points are used to identify physical noise processes that lead to a performance gain and a performance loss, respectively, in the presence of RC. The magnitude of performance gains and losses are measured by the ratio of logical error rates in the non-RC and RC settings, i.e., ${\delta }_1$ for level-1 concatenated Steane code in Fig. 7, and ${\delta }_2$ for level-2 in Fig. 7.

Figure 8

Figure 8 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. [1]. Twirling gates are inserted in Fig. 8 to tailor the noise processes to Pauli errors. These gates are compiled into existing gates by replacing easy gates by their dressed versions, in Fig. 8. The compiled circuit contains modified $\mathsf{QEC}$ routines with the twirling operators compiled into them.