Robustness of the concatenated quantum error-correction protocol against noise for channels affected by fluctuation

In quantum error correction, the description of the noise channel cannot be completely accurate and fluctuation always appears in the noise channel. It is found that when fluctuation of the physical noise channel is considered, the average effective channel is dependent only on the average of the physical noise channel, and the average of the physical noise channel here plays the role of the independent error model in previous works. Now one may conclude that in the independent error model, the results in previous works are also valid for the average channel where fluctuation exists. In some typical cases, our numerical simulations in the concatenated quantum error-correction protocol with a five-qubit code, seven-qubit Steane code, and nine-qubit Shor confirm this conjecture. For a five-qubit code, the effective channels approximate to the depolarizing channel as the concatenated level increases. For the Steane code, the effective channels approximate to one Pauli channel as the concatenated level increases. For the Shor code, the effective channels approximate to either a Pauli-$X$ or Pauli-$Z$ channel in each level, and in the next concatenated level, the effective channels approximate to the other. The numerical results for these codes indicate that the degree of approximation increases as the concatenated level increases, and the fluctuation of the noise channel decays exponentially as concatenated quantum error correction is performed. On the error-correction threshold, the attenuation ratio of the standard deviation of channel fidelity has a roughly stable value. In contrast, standard deviations of off-diagonal elements of the quantum process matrix (Pauli form) decay more quickly than standard deviations of diagonal elements.

Figure 1

(a) The way to construct the effective channel from the standard QEC protocol includes encoding, noise evolution, recovery, and decoding. (b) Our protocol where a unitary transformation $\mathcal{U}$ is sufficient to correct the errors of the principle system. The errors of the assistant qubits are left uncorrected.

Figure 2

Effective quantum process with QEC.

Figure 3

One-level QEC is performed, with $\mathcal{N}$ the amplitude damping noise. (a) Numerical results of the attenuation ratio of the SD of channel fidelity in Eq. (31) ($l=1$). The channel fidelity $f$ is set equal to 0.94, 0.95, 0.96, 0.97, and 0.98. The proportionality constant $k$ increases from 0.01 to 0.06 in steps of 0.01. (b) Numerical results of the SD of channel fidelity in Eq. (30) ($l=0,1$). The channel fidelity $f$ is set equal to 0.94, 0.95, 0.96, 0.97, and 0.98. The proportionality constant $k$ increases from 0.01 to 0.06 in steps of 0.01. In the legend, “In” means the case without performing QEC and “Out” means the case after performing one-level QEC.

Figure 4

Here $\mathcal{N}$ is the amplitude damping channel and $F^{\mathrm{avg}\left(0\right)}$ is set equal to 0.92, 0.93, 0.94, 0.95, and 0.96. For each $F^{\mathrm{avg}\left(0\right)}$, the proportionality constant $k$ increases from 0.01 to 0.06 in steps of 0.01. Numerical results of the attenuation ratio of the SD of channel fidelity in Eq. (31) ($l=1$) are depicted after one-level QEC is performed.

Figure 5

(a) Three typical cases are considered: (i) $F^{\mathrm{avg}\left(0\right)}=0.9704$, $f=0.98$, and $k=0.02$, (ii) $F^{\mathrm{avg}\left(0\right)}=0.948$, $f=0.98$, and $k=1/15$, and (iii) $F^{\mathrm{avg}\left(0\right)}=0.918$, $f=0.94$, and $k=0.05$. For each case, three noise models (depolarizing noise, amplitude damping noise, and arbitrarily generated numerical noise) are chosen for $\mathcal{N}$. Numerical results of the attenuation ratio of the SD of channel fidelity in Eq. (31) ($l=1,2,3$) are depicted after three-level concatenated QEC is performed. (b) Average effective channel fidelity $F^{\mathrm{avg}\left(0\right)}$ set equal to 0.9704, 0.948, and 0.918. The relationship between the attenuation ratio of the SD of channel fidelity in Eq. (31) and the average effective channel fidelity in Eq. (29) is depicted after three-level concatenated QEC is performed. In each typical case, three noise models are chosen for $\mathcal{N}$, and the curves coincide with each other. For clarity, we only keep the curves when $\mathcal{N}$ is set as the amplitude damping noise.

Figure 6

Relationship between the attenuation ratio of the SD of channel fidelity in Eq. (31) and the average effective channel fidelity in Eq. (29), with $\mathcal{N}$ the amplitude damping channel. Some of the data in this figure come from Figs. 3 and 5, and we also add another 42 points for one-level QEC, where $f$ is set equal to 0.9825, 0.985, 0.9875, 0.99, 0.9925, 0.995, and 0.9975. For each $f$, the number $k$ increases from 0.01 to 0.06 in steps of 0.01.

Figure 7

Fluctuation of the channel fidelity of each concatenated level, with $\mathcal{N}$ the depolarizing channel, $f=0.98$, and $k=0.02$.

Figure 8

Exact distribution of channel fidelity of each concatenated level, with $\mathcal{N}$ the depolarizing channel, $f=0.98$, and $k=0.02$.

Figure 9

One-level QEC is performed, with $\mathcal{N}$ the amplitude damping noise. (a) Numerical results of the attenuation ratio of the SD of channel fidelity in Eq. (31) ($l=1$). The channel fidelity $f$ is set equal to 0.966, 0.972, 0.978, 0.984, and 0.99. (b) Numerical results of the SD of the channel fidelity in Eq. (30) ($l=0,1$). The channel fidelity $f$ is set equal to 0.966, 0.972, 0.978, 0.984, and 0.99. In the legend, “In” means the case without performing QEC and “Out” means the case after performing one-level QEC.

Figure 10

Here $\mathcal{N}$ is the amplitude damping channel and $F^{\mathrm{avg}\left(0\right)}$ is set equal to 0.945, 0.95, 0.955, 0.96, and 0.965. For each $F^{\mathrm{avg}\left(0\right)}$, the proportionality constant $k$ increases from 0.01 to 0.06 in steps of 0.01. The numerical results of the attenuation ratio of the SD of channel fidelity in Eq. (31) ($l=1$) are depicted after one-level QEC is performed.

Figure 11

Here $\mathcal{N}$ is the amplitude damping channel and five typical cases are considered, with $f=0.98$ and $k$ increasing from 0.02 to 0.06 in steps of 0.01 ($F^{\mathrm{avg}\left(0\right)}=0.9704$, 0.9656, 0.9608, 0.956, and 0.9512). (a) Numerical results of attenuation ratio of SD of channel fidelity in Eq. (31) ($l=1,2,3$) after three-level concatenated QEC is performed. (b) Relationship between the attenuation ratio of the SD of channel fidelity in Eq. (31) and the average effective channel fidelity in Eq. (29).

Figure 12

Relationship between the attenuation ratio of the SD of channel fidelity in Eq. (31) and the average effective channel fidelity in Eq. (29), with $\mathcal{N}$ the amplitude damping channel. Some of the data are from Figs. 9 and 11, and we also add another 42 points for one-level QEC, where $f$ is set equal to 0.9825, 0.985, 0.9875, 0.99, 0.9925, 0.995, and 0.9975. For each $f$, the number $k$ increases from 0.01 to 0.06 in steps of 0.01.

Figure 13

Three typical cases are considered, with $\mathcal{N}$ the amplitude damping channel, $f=0.98$, and $k$ increasing from 0.02 to 0.06 in steps of 0.02 ($F^{\mathrm{avg}\left(0\right)}=0.9704$, 0.9608, and 0.9512). Three-level concatenated QEC is performed. (a) Numerical results of the attenuation ratio of the SD of channel fidelity in Eq. (31) ($l=1,2,3$). (b) Relationship between the attenuation ratio of the SD of channel fidelity in Eq. (31) and the average effective channel fidelity in Eq. (29).