CSS code explained

In quantum error correction, CSS codes, named after their inventors, Robert Calderbank, Peter Shor^[1] and Andrew Steane,^[2] are a special type of stabilizer code constructed from classical codes with some special properties. An example of a CSS code is the Steane code.

Construction

Let

C _{1} and C _{2} be two (classical) [n,k _{1]} , [n,k _{2]} codes such, that C _{2} \subset C _{1} and C _{1} , C ^{\perp} _{2} both have minimal distance \geq 2 t +1 , where C ^{\perp} _{2} is the code dual to C _{2} . Then define CSS (C _{1,C} _{2)} , the CSS code of C _{1} over C _{2} as an [n,k _{1} - k _{2,} d] code, with d \geq 2 t +1 as follows:

Define for

x \in C _{1} : {{|}} x + C _{2} \rangle : = 1 / \sqrt{ {{|}} C _{2} {{|}} } \sum _{y \in C _{2}} {{|}} x + y \rangle , where + is bitwise addition modulo 2. Then CSS (C _{1,C} _{2)} is defined as \{ {{|}} x + C _{2} \rangle \mid x \in C _{1} \} .

References

Book: Nielsen. Michael A.. Michael Nielsen. Chuang. Isaac L.. Isaac Chuang. Quantum Computation and Quantum Information. Cambridge University Press. Cambridge. 2010. 2nd. 844974180. 978-1-107-00217-3.

Notes and References

1. . Good quantum error-correcting codes exist . Physical Review A . 54 . 2 . 1098–1105 . 1996 . 10.1103/PhysRevA.54.1098 . 9913578 . quant-ph/9512032. 1996PhRvA..54.1098C . 11524969 .
2. Steane . Andrew . Andrew Steane . Multiple-Particle Interference and Quantum Error Correction . Proc. R. Soc. Lond. A . 452 . 1996 . 2551–2577 . 10.1098/rspa.1996.0136 . 1954. quant-ph/9601029 . 1996RSPSA.452.2551S . 8246615 .