100 research outputs found
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
Algebraic Design and Implementation of Protograph Codes using Non-Commuting Permutation Matrices
Random lifts of graphs, or equivalently, random permutation matrices, have been used to construct good families of codes known as protograph codes. An algebraic analog of this approach was recently presented using voltage graphs, and it was shown that many existing algebraic constructions of graph-based codes that use commuting permutation matrices may be seen as special cases of voltage graph codes. Voltage graphs are graphs that have an element of a finite group assigned to each edge, and the assignment determines a specific lift of the graph. In this paper we discuss how assignments of permutation group elements to the edges of a base graph affect the properties of the lifted graph and corresponding codes, and present a construction method of LDPC code ensembles based on noncommuting permutation matrices. We also show encoder and decoder implementations for these codes
Algebraic Design and Implementation of Protograph Codes using Non-Commuting Permutation Matrices
Random lifts of graphs, or equivalently, random permutation matrices, have been used to construct good families of codes known as protograph codes. An algebraic analog of this approach was recently presented using voltage graphs, and it was shown that many existing algebraic constructions of graph-based codes that use commuting permutation matrices may be seen as special cases of voltage graph codes. Voltage graphs are graphs that have an element of a finite group assigned to each edge, and the assignment determines a specific lift of the graph. In this paper we discuss how assignments of permutation group elements to the edges of a base graph affect the properties of the lifted graph and corresponding codes, and present a construction method of LDPC code ensembles based on noncommuting permutation matrices. We also show encoder and decoder implementations for these codes
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
Entanglement-assisted Coding Theory
In this dissertation, I present a general method for studying quantum error
correction codes (QECCs). This method not only provides us an intuitive way of
understanding QECCs, but also leads to several extensions of standard QECCs,
including the operator quantum error correction (OQECC), the
entanglement-assisted quantum error correction (EAQECC). Furthermore, we can
combine both OQECC and EAQECC into a unified formalism, the
entanglement-assisted operator formalism. This provides great flexibility of
designing QECCs for different applications. Finally, I show that the
performance of quantum low-density parity-check codes will be largely improved
using entanglement-assisted formalism.Comment: PhD dissertation, 102 page
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