Generality of the concatenated five-qubit code

In this work, a quantum error correction (QEC) procedure with the concatenated five-qubit code is used to construct a near-perfect effective qubit channel (with a error below ${10}^{-5}$) from arbitrary noise channels. The exact performance of the QEC is characterized by a Choi matrix, which can be obtained via a simple and explicit protocol. In a noise model with five free parameters, our numerical results indicate that the concatenated five-qubit code is general: To construct a near-perfect effective channel from the noise channels, the necessary size of the concatenated five-qubit code depends only on the entanglement fidelity of the initial noise channels.

Figure 1

(a) The way of getting the effective channel from the standard QEC protocol including encoding, noise evolution, recovery, and decoding. (b) The two processes, $\mathcal{R}\circ {\mathcal{U}}^{†}$ and ${\mathcal{U}}^{†}\circ \tilde{\mathcal{R}}$, are equivalent. (c) When $U^{†}$ for decoding is fixed according to Eq. (6), the process $\tilde{\mathcal{R}}$ can be decomposed as ${\tilde{\mathcal{R}}}_{\mathcal{SA}}={\tilde{\mathcal{R}}}_{\mathcal{A}}\otimes {\mathbb{I}}_{\mathcal{S}}$. Certainly, it can be moved away. (d) Our protocol where the chosen unitary transformation is sufficient to correct the errors of the principle system. The errors of the ancilla system are left uncorrected.

Figure 2

The entangling fidelities of the effective channel when the five-qubit code and the seven-qubit code are applied against the amplitude damping errors. The exact function in Eq. (26) is shown with the solid line, while the one in Eq. (31) for the seven-qubit code is shown with the dashed line.

Figure 3

Numerical results for the maximum distance measure $D_l^{\max }\left(F_0\right)$ defined in Eq. (38).

Figure 4

(a) For the general noise error model, the numerical results of the minimum fidelity $F_l^{\min }\left(F_0\right)$ defined in Eq. (39) are depicted. (b) Numerical results for the entangling fidelity when the initial channels are fixed to be the depolarizing ones. The results show the generality of the five-qubit code. Though the numeral results look similar in the two graphs, they are not strictly identical. More specifically, for the case $F_0=0.9$, the entangling fidelities and minimum fidelities in each level of the concatenation given in Eq. (37) and Eq. (40), respectively, are not strictly the same.