CSS code

For the document presentation language, see CSS.

In quantum error correction, Calderbank–Shor–Steane (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. Examples of CSS codes include the Steane code, the toric code, and more general surface codes.

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 $\{|x+C_2⟩\mid x\in C_1\}$.