CSS code

Short description: Class of quantum error correcting codes

In quantum error correction, CSS codes, named after their inventors, Robert Calderbank, Peter Shor and Andrew Steane, 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 [math]\displaystyle{ C_1 }[/math] and [math]\displaystyle{ C_2 }[/math] be two (classical) [math]\displaystyle{ [n,k_1] }[/math], [math]\displaystyle{ [n,k_2] }[/math] codes such, that [math]\displaystyle{ C_2 \subset C_1 }[/math] and [math]\displaystyle{ C_1 , C_2^\perp }[/math] both have minimal distance [math]\displaystyle{ \geq 2t+1 }[/math], where [math]\displaystyle{ C_2^\perp }[/math] is the code dual to [math]\displaystyle{ C_2 }[/math]. Then define [math]\displaystyle{ \text{CSS}(C_1,C_2) }[/math], the CSS code of [math]\displaystyle{ C_1 }[/math] over [math]\displaystyle{ C_2 }[/math] as an [math]\displaystyle{ [n,k_1 - k_2, d] }[/math] code, with [math]\displaystyle{ d \geq 2t+1 }[/math] as follows:

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

References

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.