# CSS code

Short description: Class of quantum error correcting codes

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 $\displaystyle{ C_1 }$ and $\displaystyle{ C_2 }$ be two (classical) $\displaystyle{ [n,k_1] }$, $\displaystyle{ [n,k_2] }$ codes such, that $\displaystyle{ C_2 \subset C_1 }$ and $\displaystyle{ C_1 , C_2^\perp }$ both have minimal distance $\displaystyle{ \geq 2t+1 }$, where $\displaystyle{ C_2^\perp }$ is the code dual to $\displaystyle{ C_2 }$. Then define $\displaystyle{ \text{CSS}(C_1,C_2) }$, the CSS code of $\displaystyle{ C_1 }$ over $\displaystyle{ C_2 }$ as an $\displaystyle{ [n,k_1 - k_2, d] }$ code, with $\displaystyle{ d \geq 2t+1 }$ as follows:

Define for $\displaystyle{ x \in C_1 : {{|}} x + C_2 \rangle := }$ $\displaystyle{ 1 / \sqrt{ {{|}} C_2 {{|}} } }$ $\displaystyle{ \sum_{y \in C_2} {{|}} x + y \rangle }$, where $\displaystyle{ + }$ is bitwise addition modulo 2. Then $\displaystyle{ \text{CSS}(C_1,C_2) }$ is defined as $\displaystyle{ \{ {{|}} x + C_2 \rangle \mid x \in C_1 \} }$.

## References

1. Robert Calderbank and Peter Shor (1996). "Good quantum error-correcting codes exist". Physical Review A 54 (2): 1098–1105. doi:10.1103/PhysRevA.54.1098. PMID 9913578. Bibcode1996PhRvA..54.1098C.
2. Steane, Andrew (1996). "Multiple-Particle Interference and Quantum Error Correction". Proc. R. Soc. Lond. A 452 (1954): 2551–2577. doi:10.1098/rspa.1996.0136. Bibcode1996RSPSA.452.2551S.

Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information (2nd ed.). Cambridge: Cambridge University Press. ISBN 978-1-107-00217-3. OCLC 844974180.