Gnu code: Difference between revisions

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{{about|for|the open-source software development project|GNU coding standards}}

{{DISPLAYTITLE:''gnu'' code}}

In [[quantum information]], the '''''gnu'' code''' refers to a particular family of [[quantum error correction|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>|0_L0_{\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>

In [[quantum information]], the '''gnu code''' refers to a particular family of [[quantum error correction|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>|1_L1_{\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>

:<math>|0_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>|1_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>|D^m_k\rangle</math> are the [[Dicke states]] consisting of a uniform superposition of all weight-''k'' words on ''m'' qubits, e.g.

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:<math>|D^4_2\rangle = \frac{|0011\rangle + |0101\rangle + |1001\rangle + |0110\rangle + |1010\rangle + |1100\rangle}{\sqrt{6}}</math>

The real parameter <math>u = \frac{m}{gn}</math> scales the density of the code. The length <math>m = gnu</math>, hence the name of the code. For odd <math>g = n</math> and <math>u \ge 1</math>, the ''gnu'' code is capable of correcting <math>\frac{g-1}{2}</math> [[erasure error]]s,<ref name="Ouyang p. ">{{cite journal | last=Ouyang | first=Yingkai | title=Permutation-invariant quantum codes | journal=Physical Review A | publisher=American Physical Society (APS) | volume=90 | issue=6 | date=2014-12-10 | issn=1050-2947 | doi=10.1103/physreva.90.062317 | page=062317| arxiv=1302.3247 | bibcode=2014PhRvA..90f2317O | s2cid=119114455 }}</ref> or [[deletion error]]s.<ref name="Ouyang 2021">{{cite webCite arXiv| last=Ouyang | first=Yingkai | title=Permutation-invariant quantum coding for quantum deletion channels | website=arXiv.org | date=2021-02-04 | urlclass=quant-ph | eprint=https://arxiv.org/abs/2102.02494v1 | access-date=2021-02-08}}</ref>

==References==