CSS code

Template:Short description Template:For 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 $C_1$ 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 $\ge 2t+1$, 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 $d\ge 2t+1$ as follows:

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