Hard decoding algorithm for optimizing thresholds under general Markovian noise

Quantum error correction is instrumental in protecting quantum systems from noise in quantum computing and communication settings. Pauli channels can be efficiently simulated and threshold values for Pauli error rates under a variety of error-correcting codes have been obtained. However, realistic quantum systems can undergo noise processes that differ significantly from Pauli noise. In this paper, we present an efficient hard decoding algorithm for optimizing thresholds and lowering failure rates of an error-correcting code under general completely positive and trace-preserving (i.e., Markovian) noise. We use our hard decoding algorithm to study the performance of several error-correcting codes under various non-Pauli noise models by computing threshold values and failure rates for these codes. We compare the performance of our hard decoding algorithm to decoders optimized for depolarizing noise and show improvements in thresholds and reductions in failure rates by several orders of magnitude. Our hard decoding algorithm can also be adapted to take advantage of a code's non-Pauli transversal gates to further suppress noise. For example, we show that using the transversal gates of the 5-qubit code allows arbitrary rotations around certain axes to be perfectly corrected. Furthermore, we show that Pauli twirling can increase or decrease the threshold depending upon the code properties. Lastly, we show that even if the physical noise model differs slightly from the hypothesized noise model used to determine an optimized decoder, failure rates can still be reduced by applying our hard decoding algorithm.

Figure 1

Representative plot of the smallest diagonal component ${\mathcal{G}}_{\sigma ,\sigma }$ of the process matrix for a noise model parametrized by $p$. As functions of $p$, the diagonal components of the process matrix approach a step function as the number of concatenation levels approaches infinity. The threshold is the smallest value $p_{\mathrm{th}}$ such that ${lim}_{t\to \infty }{\mathcal{G}}^{\left(t\right)}\left({\mathcal{N}}_p\right)=I_4$ for all $p\le p_{\mathrm{th}}$.

Figure 2

Method for selection of recovery operations for a fixed code $C$ and noise channel $\mathcal{N}$. The iterative step that calculates the process matrix at level $t$, ${\mathcal{G}}^{\left(t\right)}\left(\mathcal{N}\right)$, is equivalent to setting the recovery maps to $R_{\mathrm{opt}}=T\left(L_g^{†}\right)R$ for all $R\in {\mathcal{R}}_{\mathrm{sym}}$, where $m\left(k\right)\mathcal{G}(\mathcal{M},R)=g$ and $T\left(L_g^{†}\right)$ denotes the transversal implementation of $L_g^{†}$. This set may not be unique; see Sec. 3b for specific examples of when this can occur. We use the $l_{\infty }$ norm, that is, the maximum of the absolute values of the entries of a matrix, to test whether the process matrix has converged. However, any matrix norm can be used instead.

Figure 3

Method for lower bounding the threshold for a one-parameter noise model ${\mathcal{N}}_p$, where $t\left(p\right)$ is the minimum number of concatenation levels required to correct ${\mathcal{N}}_p$. We begin by setting $p_{\mathrm{in}}=p_{\mathrm{sym},\mathrm{thres}}$ and $\delta p=0.01$, then repeating with the new lower bound and $\delta p=0.001$ and finally $\delta p=0.0001$. To find the threshold hypersurface for an $m$-parameter noise model, repeat this procedure for ${\mathcal{N}}_{pq}$ while iterating through a mesh of points $q$.

Figure 4

Threshold curves (a) and infidelities $r$ as functions of the dephasing parameter $\lambda$ at the first (b), (c) and third (d), (e) concatenation levels for the amplitude-phase damping channel under the $\left[\right[5,1,3\left]\right]$ and Steane codes. The Shor and surface-17 codes behave similarly to the Steane code and so their infidelity curves are not displayed. Our numerics also show that for small $p$, the 5 and $X$-Shor codes have the highest thresholds due to lowest number of qubits and asymmetry of stabilizers matching the asymmetry of the noise. The amplitude damping rate is fixed at $p=0.17$ (b), (d) and $p=0.01$ (c), (e). The symmetrized and optimized curves overlap when the optimized decoder is the symmetric decoder, however, there are many regimes where the optimized decoder improves the threshold and infidelity. For the $\left[\right[5,1,3\left]\right]$ code, thresholds using the optimized decoder increase by as much as a factor of 2.14 relative to the symmetric decoder. Infidelities are lowered by as much as two orders of magnitude. The curves for the fixed symmetric decoders are all smooth, whereas the curves for the optimized decoders have kinks corresponding to points where the decoder changes to exploit asymmetries in the noise. When $\lambda$ is large compared to $p$, the noise is primarily dephasing. The $\left[\right[5,1,3\left]\right]$ code can exploit this by correcting all single- and two-qubit $Z$ errors for the first $t$ concatenation levels, until the noise becomes unbiased [the kink in (b) corresponds to $t=1$, and the two in (c) correspond to $t=2,3]$. The $X$-Shor code can also be biased to correct Pauli-$Z$ errors as it has more $X$-type stabilizers. All codes also exhibit improved performance for large values of $p$ relative to $\lambda$, that is, when the noise is primarily amplitude damping.

Figure 5

Contour plots representing hypersurfaces of the threshold value of $\theta$ for rotations around the axis $\widehat{n}=(sin\varphi cos\gamma ,sin\varphi sin\gamma ,cos\varphi )$ for (a) the $\left[\right[5,1,3\left]\right]$ code, (c) the Steane code, and (e) the Shor code using the symmetric decoder and (b) the $\left[\right[5,1,3\left]\right]$ code, (d) the Steane code, and (f) the Shor code using optimized decoding. The optimized decoder uses transversal gates to improve the threshold, particularly when the rotation is around an eigenbasis of a transversal Clifford gate (see Table 1). In particular, the $\left[\right[5,1,3\left]\right]$ code has a transversal $\pi /3$ rotation around ${\widehat{n}}_C=(1,1,1)/\sqrt{3}$ (i.e., $\gamma =\pi /4$, $\varphi =\pi /3$), which enables the optimized decoder to correct arbitrary rotations around axes close to ${\widehat{n}}_C$, illustrated by the white circular regions in (b). For Steane's code, the lightest colored regions in (d) correspond to threshold angles ${\theta }_{\mathrm{th}}\approx 0.46$ compared to ${\theta }_{\mathrm{th}}\approx 0.24$ in (c), an improvement by almost a factor of 2. The Shor code has no transversal non-Pauli gates and so the improvements from the optimized decoder are not as substantial. However, for rotations near the $y$ axis, the Shor code outperforms the Steane code by a factor of at most 2.3.

Figure 6

In (a)–(c), infidelities $r$ of the $\left[\right[5,1,3\left]\right]$ code, Shor code, and Steane code are plotted at the first and third levels for a rotation about the $x$ axis. In (a), the optimized infidelity curves are peaked at the code's threshold value ${\theta }_{\mathrm{th}}=\pi /4$. In (b), the peaks of the optimized infidelity curves are centered slightly above the code's threshold value ${\theta }_{\mathrm{th}}=0.3396$. However, the optimized level-3 infidelity curve intersects the optimized level-1 curve at the threshold value as expected. In (c), the optimized level-1 and level-3 infidelity curves intersect at the threshold value ${\theta }_{\mathrm{th}}=0.3692$. The peaks of the infidelity curves occur at $\theta =\pi /4$ due to the code's symmetry. In (d), infidelity plots of the surface-17 at the first concatenation level are shown for a rotation about the $y$ axis, $x$ axis, and the $(1,1,1)/\sqrt{3}$ axis. It can be seen that the infidelity is lowest for rotations about the $y$ axis. In all four plots, it can be seen that applying our hard decoding algorithm reduces the infidelities by, in some cases, orders of magnitude compared to the symmetric decoder.

Figure 7

(a) Threshold curves and (b) infidelities $r$ at the first and third concatenation level (with fixed $p=0.003$) for the $\left[\right[7,1,3\left]\right]$ code. Two-qubit correlated dephasing occurs with probability $q$ and depolarizing noise occurs with probability $1-q$, with a depolarizing noise parameter $p$ [see Eqs. (45) and (46)]. For small values of $p$, the $Z$ errors arising from the two-qubit correlations dominate the noise. Applying our optimized hard decoding algorithm in this regime yields a threshold of $q_{\mathrm{th}}=0.0232$. The contribution from depolarizing noise increases with $p$ until the noise is predominantly depolarizing. In this regime, the optimized decoder implements the standard CSS decoder at all levels. When $q=0$, the noise is purely depolarizing and $p_{\mathrm{th}}=0.0908$. For all values of $p$, the threshold $q_{\mathrm{th}}$ obtained by implementing our optimized decoder is larger than the threshold obtained by implementing the symmetric decoder. The level-1 and level-3 infidelity curves intersect near the respective thresholds for $p=0.003$, namely, $q_{\mathrm{th}}=0.0153$ and $0.0220$ for the symmetric and optimized decoders, respectively.

Figure 8

Threshold curves for unitary rotations about $(\frac{1}{\sqrt{2}}sin\varphi ,\frac{1}{\sqrt{2}}sin\varphi ,cos\varphi )$ by an angle $\theta$ under three different decoding schemes for the Steane code. For all values of $\varphi$, an improvement by as much as a factor of 1.7 in the threshold ${\theta }_{\mathrm{th}}$ obtained by using our algorithm optimizing over all transversal Clifford gates can be observed relative to optimizing over all Pauli gates. The Pauli twirl reduces (increases) the threshold when optimizing over all transversal Clifford (Pauli) gates for all values of $\varphi$.

Figure 9

Averaged infidelity plots over 100 random unitary operators $U$ of the effective process matrices $\mathcal{G}\left({\mathcal{N}}_p\right)$, $\mathcal{G}\left({\mathcal{N}}_{U,p}\right)$, ${\mathcal{G}}_{U,\mathrm{sym}}$, and ${\tilde{\mathcal{G}}}_U$. The figure in (a) is obtained using the $\left[\right[5,1,3\left]\right]$ code for coherent errors using random rotation axes for each random unitary. The perturbation was chosen to have the form $f\left(\theta \right)={sin}^2\theta /10$. The figure in (b) is obtained using the Steane code for the amplitude-phase damping channel. The perturbation was chosen to have the form $f\left(\lambda \right)=\lambda /10$. In (a), the inset plot shows all infidelities on a log-log scale in the regime where $\theta$ is small. As can be seen from the figure, in the regime where $\theta \gtrsim 0.185$, the optimized recovery maps for the unperturbed channel yield a lower infidelity when applied to the perturbed channel than that from applying the symmetric decoder. For smaller rotation angles, the infidelity from ${\tilde{\mathcal{G}}}_U$ is slightly larger than the infidelity arising from ${\mathcal{G}}_{U,\mathrm{sym}}$. The two differ by at most a factor of 7 in the small-$\theta$ limit. The infidelity obtained by applying the hard decoding optimization algorithm to the unperturbed channel is lowest for all sampled values of $\theta$. In (b), it can be observed that applying the decoder chosen by our optimization algorithm for the unperturbed channel to the perturbed channel results in a lower infidelity than applying the symmetric decoder to the perturbed channel, for all sampled values of $\lambda$. This indicates that our decoding scheme is very robust to small perturbations of the amplitude-phase damping channel.