Inquantum information, thegnu code refers to a particular family ofquantum error correcting codes, with the special property of being invariant underpermutations of the qubits. Given integersg (thegap),n (the occupancy), andm (the length of the code), the two codewords are

${\displaystyle |0_{\mathrm{L}}⟩=\sum _{\genfrac{}{}{0ex}{}{\ell \text{even}}{0\le \ell \le n}}\sqrt{\frac{(\genfrac{}{}{0ex}{}{n}{\ell })}{2^{n-1}}}|D_{g\ell }^m⟩}$

${\displaystyle |1_{\mathrm{L}}⟩=\sum _{\genfrac{}{}{0ex}{}{\ell \text{odd}}{0\le \ell \le n}}\sqrt{\frac{(\genfrac{}{}{0ex}{}{n}{\ell })}{2^{n-1}}}|D_{g\ell }^m⟩}$

where${\displaystyle |D_k^m⟩}$ are theDicke states consisting of a uniform superposition of all weight-k words onm qubits, e.g.

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

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

• Quantum information science
• Fault-tolerant computer systems

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