GNU PI code[1,2] 


PI code whose codewords can be expressed as superpositions of Dicke states with coefficients are square-roots of the binomial distribution.

In terms of Dicke states, logical codewords for codes encoding a single qubit [1] are \begin{align} |\overline{\pm}\rangle = \sum_{\ell=0}^{m} \frac{(\pm 1)^\ell}{\sqrt{2^m}} \sqrt{m \choose \ell} |D^n_{g \ell}\rangle~. \tag*{(1)}\end{align} Here, \(n\) is the number of particles used for encoding \(1\) qubit, and \(g, m \leq n\) are arbitrary positive integers. Codes with higher logical dimension are developed in Ref. [2]. Each Dicke state in the code can be shifted by adding a shift \(s\) to both \(n\) and \(g\).


Depends on the family. One family which is completely symmetrized versions of Bacon-Shor codes (parameterized by \(t\)) protects against arbitrary weight-\(t\) spin errors. Additionally, codes with large enough length \((t+1)(3t+1)+t\) can approximately correct \(t\) spontaneous decay errors.


For a family of shifted gnu codes, decoding can be done using projection, probability amplitude rebalancing, and gate teleportation in time \(O(n^2)\) [3].


The degree of entanglement in (non-concatenated) GNU codes scales at most logarithmically in their distance [4; Appx. D].





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Zoo Code ID: gnu_permutation_invariant

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“GNU PI code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_gnu_permutation_invariant, title={GNU PI code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“GNU PI code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.