Figure 1

(Color online) Shannon entropy of depolarizing noise $p_X=p_Y=p_Z=p$ after variable levels of self-concatenation of the [[5,1,3]] code, for noise at the threshold $(p\sim 6.29965\%)$ as well as somewhat below $(p=6.2\%)$ and above $(p=6.4\%)$ the threshold. These values are obtained from Monte Carlo simulation, with error bars considerably smaller than the plotted symbols.Reuse & Permissions

Figure 2

(Color online) Thresholds of nonzero capacity for the depolarizing channel for 5 qubit bit flip code concatenated with $n_2$ qubit phase flip code. These values are calculated exactly.Reuse & Permissions

Figure 3

(Color online) Threshold of the nonzero capacity for independent noise $(p_X,p_Y,p_Z)=\mathbf{(}{q}_X(1-q_Z),q_X{q}_Z,q_Z(1-q_X)\mathbf{)}$. The yellow line shows the line of symmetry where $q_X=q_Z$ and the red line the hashing bound for which the noise entropy is 1. The bit flip (green line) and phase flip (blue line) bounds use the optimal bit flip and phase flip code, respectively (optimization was carried out up to $n⩽1500$). The bit flip in phase flip (purple line) and phase flip in bit flip (turquoise line) bounds use the optimal $n_1$ in $n_2$ code for $n_1<9$ and $n_2⩽219$. The bit flip and phase flip bounds cross at $q_Z=q_Z=11.21074102\%$, which is the threshold for the $n_1=7$ code when $q_X=q_Z$, and the bit flip in phase flip and phase flip in bit flip bounds cross at $q_X=q_Z=11.27458434\%$, which is the threshold for the 5 in 77 code when $q_X=q_Z$ (see Table ). The right-hand panel shows an enlargement of the crossover region around the symmetry line.Reuse & Permissions

Figure 4

(Color online) Threshold of the nonzero capacity for $(p_X,p_Y,p_Z)=(p,0,p)$ channel for 5 qubit bit flip code concatenated with a $n_2$ qubit phase flip code. These values are determined exactly.Reuse & Permissions

Figure 5

(Color online) Threshold of the nonzero capacity for two Pauli noise $(p_X,p_Y,p_Z)=(p_1,p_2,p_1)$. For each type of noise, we use the optimal $n_1$ qubit bit flip code with $n_1⩽150$. To the right of the “optimal up to here” point, a code with $n_1>150$ would be optimal. These threshold values are determined exactly.Reuse & Permissions