gnu code

In quantum information, the gnu code refers to a particular family of quantum error correcting codes, with the special property of being invariant under permutations of the qubits. Given integers g (the gap), n (the occupancy), and m (the length of the code), the two codewords are

${\displaystyle |0_{\rm {L}}\rangle =\sum _{\ell \,{\textrm {even}} \atop 0\leq \ell \leq n}{\sqrt {\frac {n \choose \ell }{2^{n-1}}}}|D_{g\ell }^{m}\rangle }$

${\displaystyle |1_{\rm {L}}\rangle =\sum _{\ell \,{\textrm {odd}} \atop 0\leq \ell \leq n}{\sqrt {\frac {n \choose \ell }{2^{n-1}}}}|D_{g\ell }^{m}\rangle }$

where ${\displaystyle |D_{k}^{m}\rangle }$ are the Dicke states consisting of a uniform superposition of all weight-k words on m qubits, e.g.

${\displaystyle |D_{2}^{4}\rangle ={\frac {|0011\rangle +|0101\rangle +|1001\rangle +|0110\rangle +|1010\rangle +|1100\rangle }{\sqrt {6}}}}$

The real parameter ${\displaystyle u={\frac {m}{gn}}}$ scales the length of the code. The number ${\displaystyle u}$ needs to be at least 1. The length ${\displaystyle m=gnu}$, hence the name of the code. The distance of the code is the minimum of ${\displaystyle g}$ and ${\displaystyle n}$. For ${\displaystyle g=n}$ and ${\displaystyle u\geq 1}$, the gnu code is capable of correcting ${\displaystyle g-1}$ erasure errors, or deletion errors. The code can also correct up to ${\displaystyle \lfloor (g-1)/2\rfloor }$ corrupted qubits from the property of the distance.