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

$|0_{\mathrm{L}}⟩=\sum _{\genfrac{}{}{0ex}{}{\ell \text{<mrow data-mjx-texclass="ORD">even</mrow>}}{0\le \ell \le n}}\sqrt{\frac{(\genfrac{}{}{0ex}{}{n}{\ell })}{2^{n-1}}}|D_{g\ell }^m⟩$

$|1_{\mathrm{L}}⟩=\sum _{\genfrac{}{}{0ex}{}{\ell \text{<mrow data-mjx-texclass="ORD">odd</mrow>}}{0\le \ell \le n}}\sqrt{\frac{(\genfrac{}{}{0ex}{}{n}{\ell })}{2^{n-1}}}|D_{g\ell }^m⟩$

where $|D_k^m⟩$ are the Dicke states consisting of a uniform superposition of all weight-k words on m qubits, e.g.

$|D_2^4⟩=\frac{|0011⟩+|0101⟩+|1001⟩+|0110⟩+|1010⟩+|1100⟩}{\sqrt{6}}$

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 $u\ge 1$, the gnu code is capable of correcting $\frac{g-1}{2}$ erasure errors, or deletion errors.