Examples of minimal-memory, non-catastrophic quantum convolutional encoders

One of the most important open questions in the theory of quantum convolutional coding is to determine a minimal-memory, non-catastrophic, polynomial-depth convolutional encoder for an arbitrary quantum convolutional code. Here, we present a technique that finds quantum convolutional encoders with such desirable properties for several example quantum convolutional codes (an exposition of our technique in full generality will appear elsewhere). We first show how to encode the well-studied Forney-Grassl-Guha (FGG) code with an encoder that exploits just one memory qubit (the former Grassl-Roetteler encoder requires 15 memory qubits). We then show how our technique can find an online decoder corresponding to this encoder, and we also detail the operation of our technique on a different example of a quantum convolutional code. Finally, the reduction in memory for the FGG encoder makes it feasible to simulate the performance of a quantum turbo code employing it, and we present the results of such simulations.Comment: 5 pages, 2 figures, Accepted for the International Symposium on Information Theory 2011 (ISIT 2011), St. Petersburg, Russia; v2 has minor change

EXIT-chart aided near-capacity quantum turbo code design

High detection complexity is the main impediment in future Gigabit-wireless systems. However, a quantum-based detector is capable of simultaneously detecting hundreds of user signals by virtue of its inherent parallel nature. This in turn requires near-capacity quantum error correction codes for protecting the constituent qubits of the quantum detector against the undesirable environmental decoherence. In this quest, we appropriately adapt the conventional non-binary EXtrinsic Information Transfer (EXIT) charts for quantum turbo codes by exploiting the intrinsic quantum-to-classical isomorphism. The EXIT chart analysis not only allows us to dispense with the time-consuming Monte-Carlo simulations, but also facilitates the design of near-capacity codes without resorting to the analysis of their distance spectra. We have demonstrated that our EXIT chart predictions are in line with the Monte-Carlo simulations results. We have also optimized the entanglement-assisted QTC using EXIT charts, which outperforms the existing distance spectra based QTCs. More explicitly, the performance of our optimized QTC is as close as 0.3 dB to the corresponding hashing bound

Duality in Entanglement-Assisted Quantum Error Correction

The dual of an entanglement-assisted quantum error-correcting (EAQEC) code is defined from the orthogonal group of a simplified stabilizer group. From the Poisson summation formula, this duality leads to the MacWilliams identities and linear programming bounds for EAQEC codes. We establish a table of upper and lower bounds on the minimum distance of any maximal-entanglement EAQEC code with length up to 15 channel qubits.Comment: This paper is a compact version of arXiv:1010.550

Recursive quantum convolutional encoders are catastrophic: A simple proof

Poulin, Tillich, and Ollivier discovered an important separation between the classical and quantum theories of convolutional coding, by proving that a quantum convolutional encoder cannot be both non-catastrophic and recursive. Non-catastrophicity is desirable so that an iterative decoding algorithm converges when decoding a quantum turbo code whose constituents are quantum convolutional codes, and recursiveness is as well so that a quantum turbo code has a minimum distance growing nearly linearly with the length of the code, respectively. Their proof of the aforementioned theorem was admittedly "rather involved," and as such, it has been desirable since their result to find a simpler proof. In this paper, we furnish a proof that is arguably simpler. Our approach is group-theoretic---we show that the subgroup of memory states that are part of a zero physical-weight cycle of a quantum convolutional encoder is equivalent to the centralizer of its "finite-memory" subgroup (the subgroup of memory states which eventually reach the identity memory state by identity operator inputs for the information qubits and identity or Pauli-Z operator inputs for the ancilla qubits). After proving that this symmetry holds for any quantum convolutional encoder, it easily follows that an encoder is non-recursive if it is non-catastrophic. Our proof also illuminates why this no-go theorem does not apply to entanglement-assisted quantum convolutional encoders---the introduction of shared entanglement as a resource allows the above symmetry to be broken.Comment: 15 pages, 1 figure. v2: accepted into IEEE Transactions on Information Theory with minor modifications. arXiv admin note: text overlap with arXiv:1105.064

The Road From Classical to Quantum Codes: A Hashing Bound Approaching Design Procedure

Powerful Quantum Error Correction Codes (QECCs) are required for stabilizing and protecting fragile qubits against the undesirable effects of quantum decoherence. Similar to classical codes, hashing bound approaching QECCs may be designed by exploiting a concatenated code structure, which invokes iterative decoding. Therefore, in this paper we provide an extensive step-by-step tutorial for designing EXtrinsic Information Transfer (EXIT) chart aided concatenated quantum codes based on the underlying quantum-to-classical isomorphism. These design lessons are then exemplified in the context of our proposed Quantum Irregular Convolutional Code (QIRCC), which constitutes the outer component of a concatenated quantum code. The proposed QIRCC can be dynamically adapted to match any given inner code using EXIT charts, hence achieving a performance close to the hashing bound. It is demonstrated that our QIRCC-based optimized design is capable of operating within 0.4 dB of the noise limit

EXIT-chart aided code design for symbol-based entanglement-assisted classical communication over quantum channels

Quantum-based transmission is an attractive solution conceived for achieving absolute security. In this quest, we have conceived an EXtrinsic Information Transfer (EXIT) chart aided channel code design for symbol-based entanglement-assisted classical communication over quantum depolarizing channels. Our proposed concatenated code design incorporates a Convolutional Code (CC), a symbol-based Unity Rate Code (URC) and a soft-decision aided 2-qubit Superdense Code (2SD), which is hence referred to as a CC-URC-2SD arrangement. We have optimized our design with the aid of non-binary EXIT charts. Our proposed design operates within 1 dB of the achievable capacity, providing attractive performance gains over its bit-based counterpart. Quantitatively, the bit-based scheme requires 60% more iterations than our symbol-based scheme for the sake of achieving perfect decoding convergence. Furthermore, we demonstrate that the decoding complexity can be reduced by using memory-2 and memory-3 convolutional codes, while still outperforming the bit-based approach<br/

EXIT-chart aided quantum code design improves the normalised throughput of realistic quantum devices

In this contribution, the Hashing bound of Entanglement Assisted Quantum Channels (EAQC) is investigated in the context of quantum devices built from a range of popular materials, such as trapped ion and relying on solid state Nuclear Magnetic Resonance (NMR), which can be modelled as a so-called asymmetric channel. Then, Quantum Error Correction Codes (QECC) are designed based on Extrinsic Information Transfer (EXIT) charts for improving performance when employing these quantum devices. The results are also verified by simulations. Our QECC schemes are capable of operating close to the corresponding Hashing bound

The MacWilliams identity for quantum convolutional codes

In this paper, we propose a definition of the dual code of a quantum convolutional code, with or without entanglement assistance. We then derive a MacWilliams identity for quantum convolutional codes. Along the way, we obtain a direct proof of the MacWilliams identity, first found by Gluesing-Luerssen and Schneider, in the setting of classical convolutional codes. © 2014 IEEE

Fully-parallel quantum turbo decoder

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 dependencies 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