Effective domain

In quantum error correction, CSS codes, named after their inventors, Robert Calderbank, Peter Shor and Andrew Steane, are a special type of Stabilizer codes constructed from classical codes with some special properties.

Construction

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

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

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro. Michael A. Nielsen and Isaac L. Chuang (2010). "Quantum Computation and Quantum Information (10. Anniversary Edition)". Cambridge University Press.