Entanglement-assisted quantum turbo codes

An unexpected breakdown in the existing theory of quantum serial turbo coding is that a quantum convolutional encoder cannot simultaneously be recursive and non-catastrophic. These properties are essential for quantum turbo code families to have a minimum distance growing with blocklength and for their iterative decoding algorithm to converge, respectively. Here, we show that the entanglement-assisted paradigm simplifies the theory of quantum turbo codes, in the sense that an entanglement-assisted quantum (EAQ) convolutional encoder can possess both of the aforementioned desirable properties. We give several examples of EAQ convolutional encoders that are both recursive and non-catastrophic and detail their relevant parameters. We then modify the quantum turbo decoding algorithm of Poulin et al., in order to have the constituent decoders pass along only "extrinsic information" to each other rather than a posteriori probabilities as in the decoder of Poulin et al., and this leads to a significant improvement in the performance of unassisted quantum turbo codes. Other simulation results indicate that entanglement-assisted turbo codes can operate reliably in a noise regime 4.73 dB beyond that of standard quantum turbo codes, when used on a memoryless depolarizing channel. Furthermore, several of our quantum turbo codes are within 1 dB or less of their hashing limits, so that the performance of quantum turbo codes is now on par with that of classical turbo codes. Finally, we prove that entanglement is the resource that enables a convolutional encoder to be both non-catastrophic and recursive because an encoder acting on only information qubits, classical bits, gauge qubits, and ancilla qubits cannot simultaneously satisfy them.Comment: 31 pages, software for simulating EA turbo codes is available at http://code.google.com/p/ea-turbo/ and a presentation is available at http://markwilde.com/publications/10-10-EA-Turbo.ppt ; v2, revisions based on feedback from journal; v3, modification of the quantum turbo decoding algorithm that leads to improved performance over results in v2 and the results of Poulin et al. in arXiv:0712.288

Entanglement-assisted quantum turbo codes

An unexpected breakdown in the existing theory of quantum serial turbo coding is that a quantum convolutional encoder cannot simultaneously be recursive and non-catastrophic. These properties are essential for quantum turbo code families to have a minimum distance growing with blocklength and for their iterative decoding algorithm to converge, respectively. Here, we show that the entanglement-assisted paradigm simplifies the theory of quantum turbo codes, in the sense that an entanglement-assisted quantum (EAQ) convolutional encoder can possess both of the aforementioned desirable properties. We give several examples of EAQ convolutional encoders that are both recursive and non-catastrophic and detail their relevant parameters. We then modify the quantum turbo decoding algorithm of Poulin , in order to have the constituent decoders pass along only extrinsic information to each other rather than a posteriori probabilities as in the decoder of Poulin , and this leads to a significant improvement in the performance of unassisted quantum turbo codes. Other simulation results indicate that entanglement-assisted turbo codes can operate reliably in a noise regime 4.73 dB beyond that of standard quantum turbo codes, when used on a memoryless depolarizing channel. Furthermore, several of our quantum turbo codes are within 1 dB or less of their hashing limits, so that the performance of quantum turbo codes is now on par with that of classical turbo codes. Finally, we prove that entanglement is the resource that enables a convolutional encoder to be both non-catastrophic and recursive because an encoder acting on only information qubits, classical bits, gauge qubits, and ancilla qubits cannot simultaneously satisfy them. © 1963-2012 IEEE

Minimal realizations of linear systems: The "shortest basis" approach

Given a controllable discrete-time linear system C, a shortest basis for C is a set of linearly independent generators for C with the least possible lengths. A basis B is a shortest basis if and only if it has the predictable span property (i.e., has the predictable delay and degree properties, and is non-catastrophic), or alternatively if and only if it has the subsystem basis property (for any interval J, the generators in B whose span is in J is a basis for the subsystem C_J). The dimensions of the minimal state spaces and minimal transition spaces of C are simply the numbers of generators in a shortest basis B that are active at any given state or symbol time, respectively. A minimal linear realization for C in controller canonical form follows directly from a shortest basis for C, and a minimal linear realization for C in observer canonical form follows directly from a shortest basis for the orthogonal system C^\perp. This approach seems conceptually simpler than that of classical minimal realization theory.Comment: 20 pages. Final version, to appear in special issue of IEEE Transactions on Information Theory on "Facets of coding theory: From algorithms to networks," dedicated to Ralf Koette

Codes convolutionnels à temps variant quasi apériodiques

Éléments de la théorie du codage de canal -- Canal AWGN -- Codage aléatoire -- Principales techniques de codage -- Codes en blocs linéaires -- Codes convolutionnels -- Codes en graphes -- Codes convolutionnels à variant quasi apériodiques -- Algorithme de Viterbi adaptatif -- Analyse des codes CTV-QA systématique -- Codeurs catastrophiques -- Analyse des codes CTV-QA non systématiques -- Introduction aux codes convolutionnels à temps variant quasi apériodiques récursifs systématiques -- Propriétés des CTV-QA-RS -- Simulation des codes CTV-QA-RS de taux de codage 1/2

Τεχνικές Κωδικοποίησης Καναλιού με Έμφαση στους Συγκεραστικούς και στους Τούρμπο Κώδικες

Στην παρούσα διατριβή προτείνεται η σχεδίαση συγκεκριμένης κατηγορίας κωδίκων χαμηλής πολυπλοκότητας με κατάλληλη προσαρμογή του διαγράμματος trellis των διάτρητων συγκεραστικών κωδίκων. Στόχος είναι η βελτίωση της απόδοσης, με λογική αύξηση της πολυπλοκότητας του διαγράμματος trellis. Στα πλαίσια της έρευνας παρέχεται ένας ικανοποιητικός αριθμός νέων κωδίκων διαφόρων ρυθμών και τιμών πολυπλοκότητας. Σε πολλές περιπτώσεις διαπιστώνεται πως ελάχιστη αύξηση της πολυπλοκότητας μπορεί να οδηγήσει σε μεγάλη βελτίωση της απόδοσης, συγκριτικά με τους διάτρητους συγκεραστικούς κώδικες. Παρουσιάζεται επίσης μια μέθοδος σχεδίασης νέων ευέλικτων συγκεραστικών κωδίκων, συνδυάζοντας τις τεχνικές της απαλοιφής μονοπατιών του διαγράμματος trellis και της απαλοιφής κωδικών bit. Οι νέοι κώδικες μπορούν να μεταβάλλουν το ρυθμό τους και την πολυπλοκότητα του διαγράμματος trellis, και κατ' επέκτασιν την υπολογιστική πολυπλοκότητα της διαδικασίας αποκωδικοποίησης, οδηγώντας σε σχήματα κωδικοποίησης που κάνουν αποδοτικότερη διαχείριση των πόρων του συστήματος, εν συγκρίσει με τους κλασικούς συγκεραστικούς κώδικες μεταβλητού ρυθμού. Τέλος, εξετάζεται η δυνατότητα εφαρμογής των προαναφερθέντων αποτελεσμάτων χρησιμοποιώντας αναδρομικούς συγκεραστικούς κωδικοποιητές, οι οποίοι είναι κατάλληλοι ως περιεχόμενοι κωδικοποιητές των τούρμπο κωδίκων. Στόχος είναι η σχεδίαση αποδοτικών ευέλικτων τούρμπο σχημάτων κωδικοποίησης. Προσομοιώσεις δείχνουν ότι σε συγκεκριμένες περιοχές τιμών του σηματοθορυβικού λόγου, μια σημαντική μείωση της υπολογιστικής πολυπλοκότητας της αποκωδικοποίησης μπορεί ακόμα και να μειώσει το ρυθμό εσφαλμένων bit.In this thesis, a family of low complexity convolutional codes is constructed, by modifying appropriately the trellis diagram of punctured convolutional codes. The goal is to improve performance at the expense of a reasonable low increase of the trellis complexity. Many new convolutional codes of various code rates and values of complexity are provided. In many cases, a small increase in complexity can lead to a great improvement of performance, compared to punctured convolutional codes. Furthermore, a method is presented for designing new flexible convolutional codes, by combining the techniques of path pruning and puncturing. The new codes can vary their rate, as well as the complexity of their trellis diagram, and hence the computational complexity of the decoding algorithm, leading to coding schemes that manage more efficiently the system resources, compared to variable rate convolutional codes. Finally, the possibility of applying the aforementioned results using recursive convolutional encoders, which are used as constituent encoders in turbo codes, is investigated. The goal is to construct flexible turbo coding schemes. Simulation results indicate that in specific ranges of the SNR, a decrease in the computational complexity of the decoding procedure can even result to a decrease in the bit error rate