Quantum turbo codes (QTCs) are known to operate close to the achievable Hashing bound. However, the sequential nature of the conventional quantum turbo decoding algorithm imposes a high decoding latency, which increases linearly with the frame length. This posses a potential threat to quantum systems having short coherence times. In this context, we conceive a fully-parallel quantum turbo decoder (FPQTD), which eliminates the inherent time dependences of the conventional decoder by executing all the associated processes concurrently. Due to its parallel nature, the proposed FPQTD reduces the decoding times by several orders of magnitude, while maintaining the same performance. We have also demonstrated the significance of employing an odd-even interleaver design in conjunction with the proposed FPQTD. More specifically, it is shown that an odd-even interleaver reduces the computational complexity by 50%, without compromising the achievable performance.
Quantum turbo codes (QTCs) are known to operate close to the achievable Hashing bound. However, the sequential nature of the conventional quantum turbo decoding algorithm imposes a high decoding latency, which increases linearly with the frame length. This posses a potential threat to quantum systems having short coherence times. In this context, we conceive a fully-parallel quantum turbo decoder (FPQTD), which eliminates the inherent time dependences of the conventional decoder by executing all the associated processes concurrently. Due to its parallel nature, the proposed FPQTD reduces the decoding times by several orders of magnitude, while maintaining the same performance. We have also demonstrated the significance of employing an odd-even interleaver design in conjunction with the proposed FPQTD. More specifically, it is shown that an odd-even interleaver reduces the computational complexity by 50%, without compromising the achievable performance.
Quantum turbo codes (QTC) have shown excellent error correction capabilities in the setting of quantum communication, achieving a performance less than 1 dB away from their corresponding hashing bounds. Existing QTCs have been constructed using uniform random interleavers. However, interleaver design plays an important role in the optimization of classical turbo codes. Consequently, inspired by the widely used classical-to-quantum isomorphism, this paper studies the integration of classical interleaving design methods into the paradigm of quantum turbo coding. Simulations results demonstrate that error floors in QTCs can be lowered significantly, while decreasing memory consumption, by proper interleaving design without increasing the overall decoding complexity of the system.
Inspired by the success of classical turbo codes, quantum turbo codes (QTCs) have also been conceived for near-hashing-bound transmission of quantum information over memoryless quantum channels. However, in real physical situations, the memoryless channel assumption may not be well justified, since the channel often exhibits memory of previous error events. Here, we investigate the performance of QTCs over depolarizing channels exhibiting memory and we show that they suffer from a performance degradation at low depolarizing probability values. In order to circumvent the performance degradation issue, we conceive a new coding scheme termed quantum turbo coding scheme exploiting error-correlation (QTC-EEC) that is capable of utilizing the error-correlation while performing the iterative decoding at the receiver. The proposed QTC-EEC can achieve convergence threshold at a higher depolarizing probability for channels with a higher value of correlation parameter and achieve performance near to the capacity. Finally, we propose a joint decoding and estimation scheme for our QTC-EEC relying on the correlation estimation (QTC-EEC-E) designed for more realistic quantum systems with unknown correlation parameter. Simulation results reveal that the proposed QTC-EEC-E can achieve the same performance as that of the ideal system of known correlation parameter and hence demonstrate the accurate estimation of the proposed QTC-EEC-E.
Quantum stabilizer codes (QSCs) suffer from a low quantum coding rate since they have to recover the quantum bits (qubits) in the face of both bit-flip and phase-flip errors. In this treatise, we conceive a low-complexity concatenated quantum turbo code (QTC) design exhibiting a high quantum coding rate. The high quantum coding rate is achieved by combining the quantum-domain version of short-block codes (SBCs) also known as single parity check (SPC) codes as the outer codes and quantum unity-rate codes (QURCs) as the inner codes. Despite its design simplicity, the proposed QTC yields a near-hashing-bound error correction performance. For instance, compared to the best half-rate QTC known in the literature, namely the QIrCC-QURC scheme, which operates at the distance of D = 0.037 from the quantum hashing bound, our novel QSBC-QURC scheme can operate at the distance of D = 0.029. In addition, it is worth mentioning that this is the first instantiation of QTCs capable of adjusting the quantum encoders according to the quantum coding rate required for mitigating the Pauli errors, given the different depolarizing probabilities of the quantum channel.
Quantum turbo codes (QTC) have shown excellent error correction capabilities in the setting of quantum communication, achieving a performance less than 1 dB away from their corresponding hashing bounds. Decoding for QTCs typically assumes that perfect knowledge about the channel is available at the decoder. However, in realistic systems, such information must be estimated, and thus, there exists a mismatch between the true channel information and the estimated one. In this article, we first heuristically study the sensitivity of QTCs to such mismatch. Then, existing estimation protocols for the depolarizing channel are presented and applied in an off-line manner to provide bounds on how the use of off-line estimation techniques affects the error correction capabilities of QTCs. Finally, we present an on-line estimation method for the depolarizing probability, which, different from off-line estimation techniques, neither requires extra qubits, nor increases the latency. The application of the proposed method results in a performance similar to that obtained with QTCs using perfect channel information, while requiring less stringent conditions on the variability of the channel than off-line estimation techniques.
We present a theory of quantum serial turbo-codes, describe their iterative decoding algorithm, and study their performances numerically on a depolarization channel. Our construction offers several advantages over quantum LDPC codes. First, the Tanner graph used for decoding is free of 4-cycles that deteriorate the performances of iterative decoding. Secondly, the iterative decoder makes explicit use of the code's degeneracy. Finally, there is complete freedom in the code design in terms of length, rate, memory size, and interleaver choice. We define a quantum analogue of a state diagram that provides an efficient way to verify the properties of a quantum convolutional code, and in particular its recursiveness and the presence of catastrophic error propagation. We prove that all recursive quantum convolutional encoder have catastrophic error propagation. In our constructions, the convolutional codes have thus been chosen to be non-catastrophic and non-recursive. While the resulting families of turbo-codes have bounded minimum distance, from a pragmatic point of view the effective minimum distances of the codes that we have simulated are large enough not to degrade the iterative decoding performance up to reasonable word error rates and block sizes. With well chosen constituent convolutional codes, we observe an important reduction of the word error rate as the code length increases.
In this paper, we present a theory of quantum serial turbo codes, describe their iterative decoding algorithm, and study their performances numerically on a depolarization channel. Our construction offers several advantages over quantum low-density parity-check (LDPC) codes. First, the Tanner graph used for decoding is free of 4-cycles that deteriorate the performances of iterative decoding. Second, the iterative decoder makes explicit use of the code's degeneracy. Finally, there is complete freedom in the code design in terms of length, rate, memory size, and interleaver choice. We define a quantum analogue of a state diagram that provides an efficient way to verify the properties of a quantum convolutional code, and in particular, its recursiveness and the presence of catastrophic error propagation. We prove that all recursive quantum convolutional encoders have catastrophic error propagation. In our constructions, the convolutional codes have thus been chosen to be noncatastrophic and nonrecursive. While the resulting families of turbo codes have bounded minimum distance, from a pragmatic point of view, the effective minimum distances of the codes that we have simulated are large enough not to degrade the iterative decoding performance up to reasonable word error rates and block sizes. With well-chosen constituent convolutional codes, we observe an important reduction of the word error rate as the code length increases.