# On the Performance of Interleavers for Quantum Turbo Codes
## 2 Classical Interleavers for Quantum Turbo Codes
### Considered Classical Interleavers  
All the QTC schemes in References [3; 4; 5; 6; 7; 8; 9] are based on _random interleavers_, that is, the interleaving patterns \(\pi\) are selected at random. However, it is known from classical turbo codes that the usage of interleavers with some added structure is beneficial to reduce either the error floors or the memory requirements [11; 12; 13; 14; 15; 16; 17; 18]. In order to show that interleaver design is also beneficial when implementing quantum turbo codes, we consider the following three types of classical interleavers.  
The first classical interleavers considered here are the \(S\)-_random_ interleavers [11]. They are randomly generated by imposing the following condition on the interleaving distance or spread (\(S\)):  
\[|\pi(i)-\pi(j)|>S\text{ for }i\text{ and }j\text{ with }|i-j|\leq S. \tag{4}\]  
If the heuristic recommendation \(S<\sqrt{\frac{N}{2}}\) is satisfied, where \(N\) is the blocklength, \(S\)-random interleavers can usually be produced in reasonable time by repeatedly generating random integers until condition (4) is satisfied [11; 15]. However, as indicated in Reference [18], reasonable values of  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Config.** & **Encoder** & **R** & **E** & **m** & **Seed Transformation** \(\mathcal{U}\) \\ \hline \multirow{3}{*}{EXIT-optimized} & Outer & 1/3 & 0 & 3 & \(\{1048,3872,3485,2054,983,3164,3145,1824,987,3282,2505,1984\}_{10}\) \\ \cline{2-7}  & Inner & 1/3 & 2/3 & 3 & \(\{4091,3736,2097,1336,1601,279,3093,502,1792,3020,226,1100\}_{10}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of the Quantum Convolutional Codes (QCC) encoders. \(R\) and \(E\) refer to the coding rate and the entanglement consumption rate, respectively. \(m\) refers to the memory qubits. The seed transformations \(\mathcal{U}\) are represented using the decimal representation presented in Reference [6].  
are sometimes lower when \(N\) is large. We next consider the deterministic _Welch-Costas_[17] interleaver, which is defined by two parameters: \(p=N+1\), a prime number, and \(\alpha\), a primitive element modulo \(p\). Once \(\alpha\) is chosen for the selected blocklength, it is proven in Reference [17] that  
\[\pi(i)=(\alpha^{i}\bmod(N+1))-1,\quad\forall i\in\{0,\cdots,N-1\} \tag{5}\]  
forms a permutation which is selected as the interleaving rule. Although the spreads obtained with Welch-Costas interleavers are low, their dispersion parameter, \(\gamma\), is maximized. The dispersion of an interleaver \(\pi\) is a parameter that represents the randomness of the permutation, and it is calculated as [17]  
\[\gamma=\frac{|D(\pi)|}{\binom{N}{2}}, \tag{6}\]  
where \(|\cdot|\) indicates cardinality and \(D(\pi)\) is the set of _displacement vectors_ of \(\pi\) defined as  
\[D(\pi)=\{(j-i,\pi(j)-\pi(i))|0\leq i<j<N\}. \tag{7}\]  
The final classical interleaver we will considered for quantum error correction is the _JPL_ interleaver [17], recommended in the \"CCSDS Recommendation for Telemetry Channel Coding\" standard [19] because it has good spreading and dispersion parameters. The permutations that define this family of interleavers are constructed by applying the following algorithm:  
* Factorize the length of the interleaver \(N=k_{1}k_{2}\), where \(k_{1}=8\) usually.
* For \(s=1\) to \(s=N\) do
* \(m=(s-1)\bmod 2\)
* \(q=t\bmod 8+1\)
* \(c=(p_{q})+21m)\bmod k_{2}\)  
where \(p_{q}\) is defined as the primes \(p_{1}=31,p_{2}=37,p_{3}=43,p_{4}=47,p_{5}=53,p_{6}=59,p_{7}=61,p_{8}=67\).