gnucode

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

Inquantum information,thegnucoderefers to a particular family ofquantum error correcting codes,with the special property of being invariant underpermutationsof the qubits. Given integersg(thegap),n(the occupancy), andm(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 theDicke statesconsisting of a uniform superposition of all weight-kwords onmqubits, 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 density of the code. The length$m=gnu$,hence the name of the code. For odd$g=n$and$u\geq 1$,thegnucode is capable of correcting${\frac {g-1}{2}}$erasure errors,ordeletion errors.