In quantum error correction, 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. An example of a CSS code is the Steane code.
Let C 1 {\displaystyle C_{1}} and C 2 {\displaystyle C_{2}} be two (classical) [ n , k 1 ] {\displaystyle [n,k_{1}]} , [ n , k 2 ] {\displaystyle [n,k_{2}]} codes such, that C 2 ⊂ ⊂ --> C 1 {\displaystyle C_{2}\subset C_{1}} and C 1 , C 2 ⊥ ⊥ --> {\displaystyle C_{1},C_{2}^{\perp }} both have minimal distance ≥ ≥ --> 2 t + 1 {\displaystyle \geq 2t+1} , where C 2 ⊥ ⊥ --> {\displaystyle C_{2}^{\perp }} is the code dual to C 2 {\displaystyle C_{2}} . Then define CSS ( C 1 , C 2 ) {\displaystyle {\text{CSS}}(C_{1},C_{2})} , the CSS code of C 1 {\displaystyle C_{1}} over C 2 {\displaystyle C_{2}} as an [ n , k 1 − − --> k 2 , d ] {\displaystyle [n,k_{1}-k_{2},d]} code, with d ≥ ≥ --> 2 t + 1 {\displaystyle d\geq 2t+1} as follows:
Define for x ∈ ∈ --> C 1 : | x + C 2 ⟩ ⟩ --> := {\displaystyle x\in C_{1}:{|}x+C_{2}\rangle :=} 1 / | C 2 | {\displaystyle 1/{\sqrt {{|}C_{2}{|}}}} ∑ ∑ --> y ∈ ∈ --> C 2 | x + y ⟩ ⟩ --> {\displaystyle \sum _{y\in C_{2}}{|}x+y\rangle } , where + {\displaystyle +} is bitwise addition modulo 2. Then CSS ( C 1 , C 2 ) {\displaystyle {\text{CSS}}(C_{1},C_{2})} is defined as { | x + C 2 ⟩ ⟩ --> ∣ ∣ --> x ∈ ∈ --> 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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