CSS code

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

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 ${\displaystyle \{{|}x+C_{2}\rangle \mid x\in C_{1}\}}$.

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.