Galois-qudit non-stabilizer code[1]


The projection onto a stabilizer code is proportional to an equal sum over all elements of the stabilizer group \(\mathsf{S}\). Non-stabilizer codes generalize stabilizer codes by modifying the code projection with elements of a subset \(\mathsf{B}\subset\mathsf{S}\) called the Fourier description (see proof of Thm. 2.7 in Ref. [1]). When \(\mathsf{B}\) is a subgroup of \(\mathsf{S}\), then the code reduces to an ordinary stabilizer code.

The following non-stabilizer codes were constructed in Ref. [1]: \(((33, 155, 3))\), \(((15, 8, 3))\), \(((n, \lceil\frac{q^n}{n(q^2-1)}\rceil,2))_{GF(q)}\) and \(((n, 1+n(q-1),2))_{GF(q)}\), where \(n\) is odd. The last code family is a Galois-qudit extension of the non-additive \(((5,6,2))\) qubit code from Ref. [2].


The encoding circuit involves the application of quantum Fourier transform.


The decoding circuit involves the application of phase estimation.



  • Galois-qudit stabilizer code — A non-stabilizer code is also a stabilizer code if its Fourier description \(\mathsf{B}\) is a subgroup of some Gottesman subgroup \(\mathsf{S}\). When \(\mathsf{B}\) is just a subset, the code is explicitly not a stabilizer code.


V. Arvind, Piyush P Kurur, and K. R. Parthasarathy, “Nonstabilizer Quantum Codes from Abelian Subgroups of the Error Group”. quant-ph/0210097
E. M. Rains et al., “A Nonadditive Quantum Code”, Physical Review Letters 79, 953 (1997). DOI; quant-ph/9703002
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Internal code ID: non_stabilizer

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“Galois-qudit non-stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_non_stabilizer, title={Galois-qudit non-stabilizer code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Galois-qudit non-stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.