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Construction

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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References

1. Robert Calderbank and Peter Shor (1996). "Good quantum error-correcting codes exist". Physical Review A. 54 (2): 1098–1105. arXiv:quant-ph/9512032. Bibcode:1996PhRvA..54.1098C. doi:10.1103/PhysRevA.54.1098. PMID 9913578. S2CID 11524969. /wiki/Robert_Calderbank ↩

2. Steane, Andrew (1996). "Multiple-Particle Interference and Quantum Error Correction". Proc. R. Soc. Lond. A. 452 (1954): 2551–2577. arXiv:quant-ph/9601029. Bibcode:1996RSPSA.452.2551S. doi:10.1098/rspa.1996.0136. S2CID 8246615. /wiki/Andrew_Steane ↩