

A090899


Number of nonisomorphic indecomposable selfdual quantum codes on n qubits.


7



1, 1, 1, 2, 4, 11, 26, 101, 440, 3132, 40457, 1274068
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OFFSET

1,4


COMMENTS

Also number of nonisomorphic indecomposable selfdual codes of Type 4^H+ and length n.
Each selfdual (additive) quantum code of length n stabilizes an essentially unique quantum state on n qubits, the 2^n coefficients of which can be assumed to take values in {0,1,1}. It also corresponds to a "quantum" set of n lines in PG(n1,2): the Grassmannian coordinates of these lines sum to zero. A related sequence is the number of nonisomorphic (possibly decomposable) selfdual quantum codes on n qubits, A094927.
Also the number of equivalence classes of connected graphs on n nodes up to sequences of local complement ation (or vertex neighborhood complementation) and isomorphism.


REFERENCES

David G. Glynn and Johannes G. Maks, The classification of selfdual quantum codes of length <= 9, preprint.
D. M. Schlingemann, Stabilizer codes can be represented as graph codes, Quant. Inf. Comp. 2, 307.


LINKS

Table of n, a(n) for n=1..12.
A. Bouchet, Graphic presentations of isotropic systems, J. Combin. Theory, Ser. B, 45, (1988), 5876.
A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane, Quantum Error Correction Via Codes Over GF(4), IEEE Trans. Inform. Theory, 44 (1998), 13691387.
Lars Eirik Danielsen, Database of SelfDual Quantum Codes.
L. E. Danielsen, T. A. Gulliver, M. G. Parker, Aperiodic Propagation Criteria for Boolean Functions, preprint, 2004.
L. E. Danielsen and M. G. Parker, On the classification of all selfdual additive codes over GF(4) of length up to 12, Journal of Combinatorial Theory, Series A, Volume 113, Issue 7, October 2006, Pages 13511367
Lars Eirik Danielsen and Matthew G. Parker, Spectral Orbits and PeaktoAverage Power Ratio of Boolean Functions with respect to the {I,H,N}^n Transform, (2005), arxiv:cs/0504102. In Sequences and Their ApplicationsSETA 2004, Lecture Notes in Computer Science, Volume 3486/2005, SpringerVerlag. [Added by N. J. A. Sloane, Jul 08 2009]
David G. Glynn and Johannes G. Maks, Quantum Error Correction Project (Aotearoa), ClassSD3.pdf.
M. Hein, J. Eisert and H. J. Briegel. Multiparty entanglement in graph states, Phys. Rev. A (3) 69 (2004), no. 6, 062311, 20 pp.
G. Nebe, E. M. Rains and N. J. A. Sloane, SelfDual Codes and Invariant Theory, Springer, Berlin, 2006.


EXAMPLE

For four qubits there are two nonisomorphic selfdual quantum codes corresponding to the complete graph and the circuit on four vertices.


CROSSREFS

Cf. A094927, A110302, A110306, A151824A151827.
Sequence in context: A340651 A123432 A151398 * A159338 A159339 A159337
Adjacent sequences: A090896 A090897 A090898 * A090900 A090901 A090902


KEYWORD

hard,nonn


AUTHOR

David G Glynn (dglynn(AT)mac.com), Feb 26 2004


EXTENSIONS

a(10)a(12) from Lars Eirik Danielsen (larsed(AT)ii.uib.no) and Matthew G. Parker (matthew(AT)ii.uib.no), Jun 17 2004


STATUS

approved



