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

[math]\displaystyle{ |0_{\rm L}\rangle = \sum_{\ell\, \textrm{even}\atop 0\le\ell\le n} \sqrt{\frac{{n\choose \ell}}{2^{n-1}}} |D^m_{g\ell}\rangle }[/math]

[math]\displaystyle{ |1_{\rm L}\rangle = \sum_{\ell\, \textrm{odd}\atop 0\le\ell\le n} \sqrt{\frac{{n\choose \ell}}{2^{n-1}}} |D^m_{g\ell}\rangle }[/math]

where [math]\displaystyle{ |D^m_k\rangle }[/math] are the Dicke states consisting of a uniform superposition of all weight-k words on m qubits, e.g.

[math]\displaystyle{ |D^4_2\rangle = \frac{|0011\rangle + |0101\rangle + |1001\rangle + |0110\rangle + |1010\rangle + |1100\rangle}{\sqrt{6}} }[/math]

The real parameter [math]\displaystyle{ u = \frac{m}{gn} }[/math] scales the density of the code. The length [math]\displaystyle{ m = gnu }[/math], hence the name of the code. For odd [math]\displaystyle{ g = n }[/math] and [math]\displaystyle{ u \ge 1 }[/math], the gnu code is capable of correcting [math]\displaystyle{ \frac{g-1}{2} }[/math] erasure errors, or deletion errors.