401 research outputs found
A Class of Quantum LDPC Codes Constructed From Finite Geometries
Low-density parity check (LDPC) codes are a significant class of classical
codes with many applications. Several good LDPC codes have been constructed
using random, algebraic, and finite geometries approaches, with containing
cycles of length at least six in their Tanner graphs. However, it is impossible
to design a self-orthogonal parity check matrix of an LDPC code without
introducing cycles of length four.
In this paper, a new class of quantum LDPC codes based on lines and points of
finite geometries is constructed. The parity check matrices of these codes are
adapted to be self-orthogonal with containing only one cycle of length four.
Also, the column and row weights, and bounds on the minimum distance of these
codes are given. As a consequence, the encoding and decoding algorithms of
these codes as well as their performance over various quantum depolarizing
channels will be investigated.Comment: 5pages, 2 figure
An Adaptive Entanglement Distillation Scheme Using Quantum Low Density Parity Check Codes
Quantum low density parity check (QLDPC) codes are useful primitives for
quantum information processing because they can be encoded and decoded
efficiently. Besides, the error correcting capability of a few QLDPC codes
exceeds the quantum Gilbert-Varshamov bound. Here, we report a numerical
performance analysis of an adaptive entanglement distillation scheme using
QLDPC codes. In particular, we find that the expected yield of our adaptive
distillation scheme to combat depolarization errors exceed that of Leung and
Shor whenever the error probability is less than about 0.07 or greater than
about 0.28. This finding illustrates the effectiveness of using QLDPC codes in
entanglement distillation.Comment: 12 pages, 6 figure
Sparse Graph Codes for Quantum Error-Correction
We present sparse graph codes appropriate for use in quantum
error-correction. Quantum error-correcting codes based on sparse graphs are of
interest for three reasons. First, the best codes currently known for classical
channels are based on sparse graphs. Second, sparse graph codes keep the number
of quantum interactions associated with the quantum error correction process
small: a constant number per quantum bit, independent of the blocklength.
Third, sparse graph codes often offer great flexibility with respect to
blocklength and rate. We believe some of the codes we present are unsurpassed
by previously published quantum error-correcting codes.Comment: Version 7.3e: 42 pages. Extended version, Feb 2004. A shortened
version was resubmitted to IEEE Transactions on Information Theory Jan 20,
200
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
New constructions of CSS codes obtained by moving to higher alphabets
We generalize a construction of non-binary quantum LDPC codes over \F_{2^m}
due to \cite{KHIS11a} and apply it in particular to toric codes. We obtain in
this way not only codes with better rates than toric codes but also improve
dramatically the performance of standard iterative decoding. Moreover, the new
codes obtained in this fashion inherit the distance properties of the
underlying toric codes and have therefore a minimum distance which grows as the
square root of the length of the code for fixed .Comment: 9 pages, 9 figures, full version of a paper submitted to the IEEE
Symposium on Information Theor
High performance entanglement-assisted quantum LDPC codes need little entanglement
Though the entanglement-assisted formalism provides a universal connection
between a classical linear code and an entanglement-assisted quantum
error-correcting code (EAQECC), the issue of maintaining large amount of pure
maximally entangled states in constructing EAQECCs is a practical obstacle to
its use. It is also conjectured that the power of entanglement-assisted
formalism to convert those good classical codes comes from massive consumption
of maximally entangled states. We show that the above conjecture is wrong by
providing families of EAQECCs with an entanglement consumption rate that
diminishes linearly as a function of the code length. Notably, two families of
EAQECCs constructed in the paper require only one copy of maximally entangled
state no matter how large the code length is. These families of EAQECCs that
are constructed from classical finite geometric LDPC codes perform very well
according to our numerical simulations. Our work indicates that EAQECCs are not
only theoretically interesting, but also physically implementable. Finally,
these high performance entanglement-assisted LDPC codes with low entanglement
consumption rates allow one to construct high-performance standard QECCs with
very similar parameters.Comment: 8 pages, 5 figures. Published versio
On Dyadic Parity Check Codes and Their Generalizations
In order to communicate information over a noisy channel, error-correcting codes can be used to ensure that small errors don’t prevent the transmission of a message. One family of codes that has been found to have good properties is low-density parity check (LDPC) codes. These are represented by sparse bipartite graphs and have low complexity graph-based decoding algorithms. Various graphical properties, such as the girth and stopping sets, influence when these algorithms might fail. Additionally, codes based on algebraically structured parity check matrices are desirable in applications due to their compact representations, practical implementation advantages, and tractable decoder performance analysis.
This dissertation focuses on codes based on parity check matrices that are dyadic, n-adic, or quasi-dyadic (QD), meaning the parity check matrix representation is block structured with dyadic matrices as blocks. Depending on the number of nonzero positions in the leading row of each block, these codes may be either low density or moderate density. Since each block is reproducible, the resulting QD codes have similar advantages to quasi-cyclic (QC) codes. We examine basic code properties of dyadic, n-adic, and QD parity check codes, including bounds on the dimension and minimum distance, cycle structure of the corresponding Tanner graph, and their possible use in quantum code constructions. We also consider the relationship between cycle codes of graphs and cycle codes of their lifts.
Advisor: Christine A. Kelle
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
- …