# On the Performance of Interleavers for Quantum Turbo Codes
## 3 Results and Discussions  
In this section we present Monte Carlo simulations to asses the performance of quantum turbo codes when using the interleavers proposed in Section 2.3. To that end, the EXIT chart optimized QCCs defined by the parameters in Table 1 have been used. The interleaver length is \(N=3000\), so that the input quantum information consists of 1000 qubits. As in Reference [7], a maximum of 15 iterations has been set for the turbo decoder. In order to perform the numerical simulations, as done in References [6; 7], an \(n\)-qubit error is randomly generated in each transmission round as explained in (1). At the decoder, the syndromes \(R_{2}\) and \(S_{1}^{x}\) are computed first, and the turbo decoding algorithm runs until the hard decisions on the estimated logical errors are the same as in the previous iteration or until the number of iterations reaches the maximum value (which we fix to 15). The operational figure of merit selected in order to evaluate the performance of these quantum error correction schemes is the _Word Error Rate_ (WER), which is the probability that at least one qubit of the received block is incorrectly decoded. Regarding the numerical Monte Carlo methods used in order to estimate the WER of the different QTCs, the following rule of thumb has been used [20] in order to select the number of blocks to be transmitted, \(N_{blocks}\):  
\[N_{blocks}=\frac{100}{WER}. \tag{8}\]  
As explained in Reference [20], and under the assumption that the observed error events are independent, this results in a 95% confidence interval of about \((0.8 WER,1.25 WER)\), where \(WER\) refers to the empirically estimated value for the WER. The following interleavers have been implemented:  
1. \(S\)-random interleaver with parameter \(S=25\),
2. Welch-Costas interleaver with parameter \(\alpha=2987\) and
3. JPL interleaver. The spread (\(S\)) and dispersion (\(\gamma\)) parameters of such designed interleavers are indicated in Table 2.  
Figure 2 shows the results of Monte Carlo simulations for the different QTC schemes in the depolarizing channel. It can be seen that the performance of all QTCs is similar in the turbo-cliff region, achieving a gap (Note that the gap is calculated in a similar fashion as in Reference [7], that is, the distance between the QTC convergence threshold \(p=0.35\) and the EA Hashing limit is taken as \(10\log_{10}(0.3779/0.35)=0.3\) dB.) to the EA Hashing limit of 0.3 dB, which is also the gap obtained by the random interleaver QTCs of Reference [7]. However, different behaviors can be observed in the error floor region. As depicted in Figure 2, both the JPL and the \(S\)-random interleavers present a much lower error floor than that of random interleaving. Specifically, the error floor of the original ramdomly interleaved QTCs is of the order of \(10^{-2}\), while the error floor of the \(S\)-random interleaver is around \(10^{-4}\) and that of the JPL is of order of \(10^{-3}\)-\(10^{-4}\). Notice that the error floor of the JPL interleaver presents a larger slope than that of the other interleavers, achieving levels close to those of the \(S\)-random interleaver when the channel quality improves. This means that quantum error correction systems using \(S\)-random and JPL interleavers present error floors around two orders of magnitude better than those obtained with random interleavers. Figure 2 shows that the performance of the Welch-Costas interleaver is similar to that of the baseline random interleaver. However, the benefit of using the Welch-Costas interleaver is that its memory requirements are much smaller, as the permutation is defined just by parameters \(N\) and \(\alpha\). On the other hand, the random (and \(S\)-random) permutations must be stored completely. This means that using the Welch-Costas interleaver is a good option for systems with strict storage requirements.