CSS code

In quantum error correction, CSS codes, named after their inventors, A. R. Calderbank, Peter Shor and Andrew Steane, are a special type of Stabilizer codes constructed from classical codes with some special properties.

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 $C_1 , C_2^\perp$ both have minimal distance $\geq 2t%2B1$ , where $C_2^\perp$ is the code dual to $C_2$ . Then define $\text{CSS}(C_1,C_2)$ , the CSS code of $C_1$ over $C_2$ as an $[n,k_1 - k_2, d]$ code, with $d \geq 2t%2B1$ as follows:

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

References

Michael A. Nielsen and Isaac L. Chuang (2010). "Quantum Computation and Quantum Information (10. Anniversary Edition)". Cambridge University Press.