Enhanced Feedback Iterative Decoding of Sparse Quantum Codes

Decoding sparse quantum codes can be accomplished by syndrome-based decoding using a belief propagation (BP) algorithm.We significantly improve this decoding scheme by developing a new feedback adjustment strategy for the standard BP algorithm. In our feedback procedure, we exploit much of the information from stabilizers, not just the syndrome but also the values of the frustrated checks on individual qubits of the code and the channel model. Furthermore we show that our decoding algorithm is superior to belief propagation algorithms using only the syndrome in the feedback procedure for all cases of the depolarizing channel. Our algorithm does not increase the measurement overhead compared to the previous method, as the extra information comes for free from the requisite stabilizer measurements.Comment: 10 pages, 11 figures, Second version, To be appeared in IEEE Transactions on Information Theor

Fifteen years of quantum LDPC coding and improved decoding strategies

The near-capacity performance of classical low-density parity check (LDPC) codes and their efficient iterative decoding makes quantum LDPC (QLPDC) codes a promising candidate for quantum error correction. In this paper, we present a comprehensive survey of QLDPC codes from the perspective of code design as well as in terms of their decoding algorithms. We also conceive a modified non-binary decoding algorithm for homogeneous Calderbank-Shor-Steane-type QLDPC codes, which is capable of alleviating the problems imposed by the unavoidable length-four cycles. Our modified decoder outperforms the state-of-the-art decoders in terms of their word error rate performance, despite imposing a reduced decoding complexity. Finally, we intricately amalgamate our modified decoder with the classic uniformly reweighted belief propagation for the sake of achieving an improved performance

Modified belief propagation decoders applied to non-CSS QLDGM codes.

Quantum technology is becoming increasingly popular, and big companies are starting to invest huge amounts of money to ensure they do not get left behind in this technological race. Presently, qubits and operational quantum channels may be thought of as far-fetched ideas, but in the future, quantum computing will be of critical importance. In this project, it is provided a concise overview of the basics of coding theory and how they can be used in the design of quantum computers. Specifically, Low Density Parity Check (LDPC) codes are focused, as they can be integrated within the stabilizer construction to build effective quantum codes. Following this, it is introduced the specifics of the quantum paradigm and present the most common family of quantum codes: stabilizer codes. Finally, it is explained the codes that have been used in this project, discussing what type of code they are and how they are designed. In this last section, it is also presented the ultimate goal of the project: using modified belief propagation decoders that had previously been tested for QLDPCs, for the proposed non-CSS QLDGM codes of this project

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

Trapping Sets of Quantum LDPC Codes

Iterative decoders for finite length quantum low-density parity-check (QLDPC) codes are attractive because their hardware complexity scales only linearly with the number of physical qubits. However, they are impacted by short cycles, detrimental graphical configurations known as trapping sets (TSs) present in a code graph as well as symmetric degeneracy of errors. These factors significantly degrade the decoder decoding probability performance and cause so-called error floor. In this paper, we establish a systematic methodology by which one can identify and classify quantum trapping sets (QTSs) according to their topological structure and decoder used. The conventional definition of a TS from classical error correction is generalized to address the syndrome decoding scenario for QLDPC codes. We show that the knowledge of QTSs can be used to design better QLDPC codes and decoders. Frame error rate improvements of two orders of magnitude in the error floor regime are demonstrated for some practical finite-length QLDPC codes without requiring any post-processing.Comment: Revised version - 19 pages, 12 figures - Accepted for publication in Quantu