Quantum stabilizer codes for correlated and asymmetric depolarizing errors

We study the performance of common quantum stabilizer codes in the presence of asymmetric and correlated errors. Specifically, we consider the depolarizing noisy quantum memory channel and perform quantum error correction via the five- and seven-qubit stabilizer codes. We characterize these codes by means of the entanglement fidelity as a function of the error probability and the degree of memory. We show that their performances are lowered by the presence of correlations, and we compute the error probability threshold values for code effectiveness. Furthermore, we uncover that the asymmetry in the error probabilities does not affect the performance of the five-qubit code, while it does affect the performance of the seven-qubit code, which results in being less effective when considering correlated and symmetric depolarizing errors but more effective for correlated and asymmetric errors.

Figure 1

Threshold curve for the five-qubit code.

Figure 2

${\mathcal{F}}^{\left[[5,1,3]\right]}(\mu ,p)$ vs $\mu$ with $0⩽\mu ⩽0.33\hspace{0.5em}$(for $\mu >0.33$, the error-correction scheme is not effective anymore) for $p=4.33\times {10}^{-2}$ (thick solid line), $p=4\times {10}^{-2}$ (thin solid line), and $p=3.67\times {10}^{-2}$ (dashed line).

Figure 3

Symmetric case: Threshold curves ${\mu }_{\mathrm{th}}^{\left[[5,1,3]\right]}\left(p\right)$ (dashed line), ${\mu }_{\mathrm{th},\text{Set-2}}^{\left[[7,1,3]\right]}\left(p\right)$ (thin solid line), and ${\mu }_{\mathrm{th},\text{Set-1}}^{\left[[7,1,3]\right]}\left(p\right)$ (thick solid line) vs $p$.

Figure 4

${\mathcal{F}}_{\text{Set-}1}^{\left[[7,1,3]\right]}(\mu ,p)$ vs $\mu$ with $0⩽\mu ⩽0.199$ (for $\mu >0.199$, the error-correction scheme is not effective anymore) for $p=4.33\times {10}^{-2}$ (thick solid line), $p=4\times {10}^{-2}$ (thin solid line), and $p=3.67\times {10}^{-2}$ (dashed line).

Figure 5

Asymmetric case: Threshold curves ${\mu }_{\mathrm{th},\text{Set-}2}^{\left[[7,1,3]\right]}\left(p\right)$ (dashed line) and ${\mu }_{\mathrm{th}}^{\left[[5,1,3]\right]}\left(p\right)$ (thin solid line) vs $p$.

Figure 6

${\mathcal{F}}_{\mathrm{asym}}^{\left[[7,1,3]\right]}(\mu ,p)$ (dashed line), ${\mathcal{F}}_{\mathrm{asym}}^{\left[[5,1,3]\right]}={\mathcal{F}}_{\mathrm{sym}}^{\left[[5,1,3]\right]}(\mu ,p)$ (thin solid line), and ${\mathcal{F}}_{\mathrm{sym}}^{\left[[7,1,3]\right]}(\mu ,p)$ (thick solid line) vs $\mu$ with $0⩽\mu \lesssim 0.199$ (where both error-correction schemes are effective) for $p=4\times {10}^{-2}$ with ${\alpha }_Z=25\alpha$, ${\alpha }_X=5\alpha$, and ${\alpha }_Y=\alpha$.