Quantum stabilizer and quantum LDPC codes

Ting-Chun Lin, Min-Hsiu Hsieh

Quantum low-density parity-check (qLDPC) codes are quantum stabilizer codes where each stabilizer acts on a constant number of qubits and each qubit is acted on by a constant number of stabilizers. We study qLDPC codes constructed from balanced products and lossless expanders. We found that assuming the existence of 2-sided lossless expander graphs with free group action, the resulting qLDPC codes have constant rate, linear distance, and linear time decoders.

Ivan B. Djordjević

Jean–Pierre Tillich

Quantum low-density parity-check (LDPC) codes are stabilizer codes in which each stabilizer generator acts on O(1) qubits and each qubit participates in O(1) generators. Some famous classes of codes such as the toric code are quantum LDPC codes. There are several reasons that explain why this subclass of stabilizer codes is particularly interesting. For instance, last year it was shown by Gottesman that such codes would in principle allow fault-tolerant quantum computing with constant overhead.However, for this application we need quantum LDPC codes that are able to encode a linear fraction of qubitswith good minimum distance properties and that are equipped with an efficient decoding algorithm.The fact that their classical counterpart enjoys all these nice properties could lead to think that it will be easyto obtain the same result in the quantum setting. However it has turned out that decoding quantum LDPC codes is a much more involved issue than decoding classical LDPC codes.The latter class of codes is decoded successfully by low complexity iterative decoding algorithms.In principle, such algorithms could also be used to decode quantum LDPC codes.However, this approach fails for quantum LDPC codes due to the fact that suchcodes are inherently highly degenerate. I will survey and discuss in this talk alternative approaches that have been successful for some subclasses of quantum LDPC codes.

Joschka Roffe

Thomas Camara, Ollivier, H., Jean–Pierre Tillich

Two methods for constructing quantum LDPC codes are presented. We explain how to overcome the difficulty of finding a set of low weight generators for the stabilizer group of the code. Both approaches are based on some graph representation of the generators of the stabilizer group and on simple local rules that ensure commutativity. A message passing algorithm for generic quantum LDPC codes is also introduced. Finally, we provide two specific examples of quantum LDPC codes of rate 1/2 obtained by our methods, together with a numerical simulation of their performance over the depolarizing channel.

Nadja Willenborg