CSS code

For the document presentation language, see CSS.

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.

Construction

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

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 . doi:10.1103/PhysRevA.54.1098.
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.

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.