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 L ⟩ = ∑ ℓ even 0 ≤ ℓ ≤ n ( n ℓ ) 2 n − 1 | D g ℓ m ⟩ {\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 }
| 1 L ⟩ = ∑ ℓ odd 0 ≤ ℓ ≤ n ( n ℓ ) 2 n − 1 | D g ℓ m ⟩ {\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 | D k m ⟩ {\displaystyle |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 ⟩ = | 0011 ⟩ + | 0101 ⟩ + | 1001 ⟩ + | 0110 ⟩ + | 1010 ⟩ + | 1100 ⟩ 6 {\displaystyle |D_{2}^{4}\rangle ={\frac {|0011\rangle +|0101\rangle +|1001\rangle +|0110\rangle +|1010\rangle +|1100\rangle }{\sqrt {6}}}}
The real parameter u = m g n {\displaystyle u={\frac {m}{gn}}} scales the length of the code. The number u {\displaystyle u} needs to be at least 1. The length m = g n u {\displaystyle m=gnu} , hence the name of the code. The distance of the code is the minimum of g {\displaystyle g} and n {\displaystyle n} . For g = n {\displaystyle g=n} and u ≥ 1 {\displaystyle u\geq 1} , the gnu code is capable of correcting g − 1 {\displaystyle g-1} erasure errors, or deletion errors. The code can also correct up to ⌊ ( g − 1 ) / 2 ⌋ {\displaystyle \lfloor (g-1)/2\rfloor } corrupted qubits from the property of the distance.
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