gnu code

For the open-source software development project, see GNU coding standards.

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

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

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

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

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

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