Error Correction using Quantum Quasi-Cyclic

Low Density Parity Check (LDPC) Codes

Lin Jing

Why use LDPC code

Classical LDPC code can approach Shannon Capacity Efficient decoders

Main Obstacle is dual containing constraint.

CSS Construction

Use C as parity check matrix for X error, and the dual code of C as parity check matrix for Z error

“Quantum Quasi-Cyclic LDPC Codes”, Manabu et. al, IEEE Intl. Symp. Inform. Theory, 2007

Generate Parity Check Matrices

A computational method:

Use this method, we can generate Hc and HD

They are dual code of each other.

1 10 27 51 72 63 46 22 29 71 53 19 44 2 20 5422 1 10 27 51 72 63 46 71 53 19 44 2 20 54 2946 22 1 10 27 51 72 63 53 19 44 2 20 54 29 7163 46 22 1 10 27 51 72 19 44 2 20 54 29 71 5372 63 46 22 1 10 27 51 44 2 20 54 29 71 53 1951 72 63 46 22 1 10 27 2 20 54 29 71 53 19 4427 51 72 63 46 22 1 10 20 54 29 71 53 19 44 210 27 51 72 63 46 22 1 54 29 71 53 19 44 2 20

44 2 20 54 29 71 53 19 72 63 46 22 1 10 27 5119 44 2 20 54 29 71 53 63 46 22 1 10 27 51 7253 19 44 2 20 54 29 71 46 22 1 10 27 51 72 6371 53 19 44 2 20 54 29 22 1 10 27 51 72 63 4629 71 53 19 44 2 20 54 1 10 27 51 72 63 46 2254 29 71 53 19 44 2 20 10 27 51 72 63 46 22 120 54 29 71 53 19 44 2 27 51 72 63 46 22 1 10 2 20 54 29 71 53 19 44 51 72 63 46 22 1 10 27

Rows: 584 Columns: 1168

Deleted rows: 146/584Code rate: 25%

Sum-Product Alg. Decoder

Use different error probability setting for decoder