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 ${\displaystyle C_{1}}${\displaystyle C_{1}} and ${\displaystyle C_{2}}${\displaystyle C_{2}} be two (classical) ${\displaystyle [n,k_{1}]}${\displaystyle [n,k_{1}]}, ${\displaystyle [n,k_{2}]}${\displaystyle [n,k_{2}]} codes such, that ${\displaystyle C_{2}\subset C_{1}}${\displaystyle C_{2}\subset C_{1}} and ${\displaystyle C_{1},C_{2}^{\perp }}${\displaystyle C_{1},C_{2}^{\perp }} both have minimal distance ${\displaystyle \geq 2t+1}${\displaystyle \geq 2t+1}, where ${\displaystyle C_{2}^{\perp }}${\displaystyle C_{2}^{\perp }} is the code dual to ${\displaystyle C_{2}}${\displaystyle C_{2}}. Then define ${\displaystyle {\text{CSS}}(C_{1},C_{2})}${\displaystyle {\text{CSS}}(C_{1},C_{2})}, the CSS code of ${\displaystyle C_{1}}${\displaystyle C_{1}} over ${\displaystyle C_{2}}${\displaystyle C_{2}} as an ${\displaystyle [n,k_{1}-k_{2},d]}${\displaystyle [n,k_{1}-k_{2},d]} code, with ${\displaystyle d\geq 2t+1}${\displaystyle d\geq 2t+1} as follows:

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

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