# On the Performance of Interleavers for Quantum Turbo Codes
## 1 Introduction  
Classical turbo codes, introduced by [PERSON] et al. in Reference [1], marked a breakthrough in classical coding theory as the first practical codes closely approaching [PERSON]'s channel capacity [2]. Motivated by their success, serial concatenation of Quantum Convolutional Codes (QCC) was proposed in Reference [3]. The use of a serial structure in Reference [3] is explained by the laws of quantum mechanics, in particular by the underlying classical-to-quantum isomorphism [4] applied in the design of quantum error correction codes (QECC): Classical turbo codes are generally systematic, interleaved parallel concatenations of convolutional codes but in the quantum domain such a systematic parallel structure is prohibited by the no-cloning theorem. The performance of the Quantum Turbo Codes (QTCs) proposed in Reference [3] is not on par with that of classical serial turbo codes. The reason is that unassisted QCCs cannot be both non-catastrophic and recursive at the same time, properties that are required for the inner code of a turbo code so that the minimum distance grows with the block length and the iterative decoding algorithm achieves convergence. Consequently, non-catastrophic, non-recursive QCCs were used in Reference [3], resulting in codes with bounded minimum distance. Utilizing the entanglement-assistance (EA) techniques (Entanglement-assistance techniques utilize entangled qubits pre-shared between the sender and the receiver, which simplifies the construction of stabilizer QECCs by relaxing the stringent commutativity conditions that the stabilizers need to satisfy.) for QECCs presented in Reference [5], this issue was addressed by [PERSON] et al. in [6], which proposed QTCs operating within 1 dB of their corresponding EA hashing bounds. Taking a step further, [PERSON] et al. proposed in Reference [7] an EXtrinsic Information Transfer (EXIT) chart technique to narrow the gap to the Hashing bound, resulting in codes with a performance as close as 0.3 dB to the hashing bounds. However, these QTC codes present error floors that are higher than in the original distance-spectra-based codes in Reference [6]. Based on the use of Quantum IrRegular Convolutional Codes (QIRCC) as outercodes, the authors in Reference [8] showed a performance in the turbo-cliff region similar to that of the EXIT optimized codes in Reference [7] but with lower error floors. An entanglement-assisted version of QIRCCs was introduced in Reference [9] with the aim of designing efficient concatenated QECCs for asymmetric depolarizing channels. Finally, a Quantum Unity Rate Code (QURC) aided QTC scheme was proposed in Reference [10] in order to improve the performance of the outer code without experiencing code rate reduction due to the inner code. The serial concatenation of the inner and outer QCCs used to construct QTCs is realized by means of an interleaver, which permutes the symbols so that the error locations are randomized and error correction can be improved. The reason for the use of an interleaver in concatenated coding schemes is that the first stage in the decoding process generates bursts of errors that are more efficiently corrected in the second stage if they are scrambled. The QTCs proposed in the literature [3; 4; 5; 6; 7; 8; 9] use the so-called random interleaver. However, it is known from classical turbo codes that interleaving design plays a central role in optimizing performance, specially when the error floor region is considered [11; 12; 13; 14; 15; 16; 17; 18]. Motivated by such studies, in this paper we investigate the application of different types of interleavers in QTCs, aiming at reducing the error floors. Simulation results show that the QTCs designed using the proposed interleavers present the same behavior in the turbo-cliff region as the codes with random interleavers in Reference [7], while the performance in the error floor region is improved by up to two orders of magnitude.