Good families of quantum low-density parity-check codes and a geometric framework for the amplitude-damping channel

Classical low-density parity-check (LDPC) codes were first introduced by Robert Gallager in the 1960's and have reemerged as one of the most influential coding schemes. We present new families of quantum low-density parity-check error-correcting codes derived from regular tessellations of Platonic 2-manifolds and from embeddings of the Lubotzky-Phillips-Sarnak Ramanujan graphs. These families of quantum error-correcting codes answer a conjecture proposed by MacKay about the existence of good families of quantum low-density parity-check codes with nonzero rate, increasing minimum distance and a practical decoder. For both families of codes, we present a logarithmic lower bound on the shortest noncontractible cycle of the tessellations and therefore on their distance. Note that a logarithmic lower bound is the best known in the theory of regular tessellations of 2-manifolds. We show their asymptotic sparsity and non-zero rate. In addition, we show their decoding performance with simulations using belief propagation. Furthermore, we present a general geometrical model to design non-additive quantum error-correcting codes for the amplitude-damping channel. Non-additive quantum error-correcting codes are more general than stabilizer or additive quantum errorcorrecting codes, and in some cases non-additive quantum codes are more optimal. As an example, we provide an 8-qubit amplitude-damping code, which can encode 1 qubit and correct for 2 errors. This violates the quantum Hamming bound which requires that its length start at 9.

Department dc:contributor.department

Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2011

Author dc:creator

• Riemann, Reina, 1975-

Advisor dc:contributor.advisor

• Peter W. Shor.

• Electrical Engineering and Computer Science.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/64585

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/64585