A090899 Number of nonisomorphic indecomposable self-dual quantum codes on n qubits.

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

Offset: 1

Author

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

Keywords

• hard
• nonn

Comments

Also number of nonisomorphic indecomposable self-dual codes of Type 4^H+ and length n.

Each self-dual (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(n-1,2): the Grassmannian coordinates of these lines sum to zero. A related sequence is the number of nonisomorphic (possibly decomposable) self-dual 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.

Examples

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

References

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

Links

• A. Bouchet, Graphic presentations of isotropic systems, J. Combin. Theory, Ser. B, 45, (1988), 58-76.
• 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), 1369-1387.
• Lars Eirik Danielsen, Database of Self-Dual 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 self-dual additive codes over GF(4) of length up to 12, Journal of Combinatorial Theory, Series A, Volume 113, Issue 7, October 2006, Pages 1351-1367
• Lars Eirik Danielsen and Matthew G. Parker, Spectral Orbits and Peak-to-Average Power Ratio of Boolean Functions with respect to the {I,H,N}^n Transform, (2005), arxiv:cs/0504102. In Sequences and Their Applications-SETA 2004, Lecture Notes in Computer Science, Volume 3486/2005, Springer-Verlag. [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. Multi-party 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, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.

Crossrefs

Cf. A094927, A110302, A110306, A151824-A151827.

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