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Permutationally invariant codes for quantum error correction

✍ Scribed by Harriet Pollatsek; Mary Beth Ruskai

Publisher
Elsevier Science
Year
2004
Tongue
English
Weight
408 KB
Volume
392
Category
Article
ISSN
0024-3795

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✦ Synopsis

A permutationally invariant n-bit code for quantum error correction can be realized as a subspace stabilized by the non-Abelian group S n . The code is spanned by bases for the trivial representation, and all other irreducible representations, both those of higher dimension and orthogonal bases for the trivial representation, are available for error correction.

A number of new (non-additive) binary codes are obtained, including two new 7-bit codes and a large family of new 9-bit codes. It is shown that the degeneracy arising from permutational symmetry facilitates the correction of certain types of two-bit errors. The correction of two-bit errors of the same type is considered in detail, but is shown not to be compatible with single-bit error correction using 9-bit codes.

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An (m) error at bit i causes bits i,i + 1,..., and i+ m -1 (or up to the end of the word) to be in error, inflicting m consecutive errors. The most practical cases are when m --2 which is referred to as adjacent errors and when m --n (the length of the word) in which an error causes the rest of the