Group theory for quantum gates and quantum coherence

Journal of Physics A: Mathematical and Theoretical

Volumn 41, Issue 18, 2008, Pages

  Planat, Michel ^a   Jorrand, Philippe ^b  

^a CNRS   (France)

^b UNIV GRENOBLE ALPES   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 42449128302     PISSN: 17518113     EISSN: 17518121     Source Type: Journal    
DOI: 10.1088/1751-8113/41/18/182001     Document Type: Article

References (39)

1. Boykin O, Mor T, Pulver M, Roychowdhury V and Vatan F 1999 On universal and fault-tolerant quantum computing: a novel basis and a new constructive proof of universality for Shor's basis 40th Annual Symposium on the Foundations of Computer Science pp 486-94 (Preprint quant-ph/9906054)
2. S Francoise J P, Naber G L and Tsou S T 2006 Quantum error correction and fault tolerance Encyclopedia of Mathematical Physics vol 4 ed D Gottesman (Oxford: Elsevier) pp 196-201 (Preprint quant-ph/0507174)
3. Gottesman D and Chuang I L 1999 Quantum teleportation is a universal computational primitive Nature 402 390-2
4. Kitaev A Yu 1997 Fault-tolerant quantum computation with anions Preprint quant-ph/9707021
5. Raussendorf R, Harrington J and Goyal K 2007 Topological fault-tolerance in cluster state quantum computation New J. Phys. 9 199
6. Wu L A, Zanardi P and Lidar D A 2005 Holonomic quantum computation in decoherence-free subspaces Phys. Rev. Lett. 95 130501
7. Perdrix S and Jorrand Ph 2006 Classical-controlled quantum computation Math. Structures Comp. Sci. 16 601-20
8. Gottesman D 1998 The Heisenberg representation of quantum computers Preprint quant-ph/9807006
9. Clark S, Jozsa R and Linden N 2008 Generalized Clifford groups and simulation of associated quantum circuits Quant. Inf. Comp. 8 106-26
10. Calderbank A R, Rains E M, Schor P W and Sloane N J A 1998 Quantum error correction via codes over GF(4) IEEE Trans. Inform. Theory 44 1369-87
11. Nielsen M A and Chuang I L 2000 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press)
12. Perdrix S 2007 Toward minimal resources of measurement-based quantum computation New J. Phys. 9 206
13. Planat M and Saniga M 2008 On the Pauli graphs of N-qudits Quant. Inf. Comp. 8 127-46
14. Planat M, Saniga M and Kibler M 2006 Quantum entanglement and projective ring geometry Sigma 2 66
15. Saniga M and Planat M 2007 Multiple qubits as symplectic polar spaces of order two Adv. Stud. Theor. Phys. 1 1-4
16. Planat M 2007 Clifford quantum computer and the Mathieu groups Preprint quant-ph/0711.1733
17. Bombin H and Martin-Delgado M A 2006 Topological quantum distillation Phys. Rev. Lett. 97 180501
18. Aerts D and Czachor M 2007 Cartoon computation: Quantum-like algorithms without quantum mechanics J. Phys. A: Math. Theor. 40 F259-66
19. Zeier R, Grassl M and Beth T 2004 Gate simulation and lower bounds on the simulation time Phys. Rev. A 70 032319
20. The GAP Group 2004 GAP-Groups, Algorithms, and Programming, Version 4.4 (http://www.gap-system.org)
21. Bosma W, Cannon J and Playoust C 1997 The Magma algebra system. I. The user language J. Symb. Comput. 24 235-65
22. Milne J S Group theory (available on line at http://www.jmilne.org/math/)
23. Kappe L C and Morse R F On commutators in groups (available on line at http://faculty.evansville.edu/ rm43/publications/commutatorsurvey.pdf)
24. Banica T, Bichon J and Collins B 2007 The hyperoctahedral quantum group Preprint math.RT/0701859
25. Klappenecker A and Rötteler M 2002 Beyond stabilizer codes: I. Nice error bases IEEE Trans. Inform. Theory 48 2392-5
26. Klappenecker A and Rötteler M 2002 Beyond stabilizer codes II: Clifford codes IEEE Trans. Inform. Theory 48 2396-9
27. ATLAS of Finite Group Representations http://brauer.maths.qmul.ac.uk/ Atlas/v3/misc/M20/
28. Wilson R A The finite simple groups (available at http://www.maths.qmul. ac.uk/∼raw/fsgs.html)
29. Preskill J 1998 Fault tolerant quantum computation Introduction to Quantum Computation and Information ed H K Lo, T Spiller and S Popescu (Singapore: World-Scientific) (Preprint quant-ph/9712048)
30. Ogburn R W and Preskill J 1999 Topological Quantum Computation (Lecture Notes in Computer Science vol 1509) pp 341-56
31. Kauffman L H and Lomonaco S J 2004 Braiding operators are universal quantum gates New J. Phys. 6 134
32. Benjamin S et al 2006 Toward a fullerene-based quantum computer J. Phys.: Condens. Matter 18 S867-83
33. Planat M, Rosu H C and Perrine S 2006 A survey of finite algebraic geometrical structures underlying mutually unbiased measurements Found. Phys. 36 1662-80
34. Planat M 2001 1/f noise, the measurement of time and number theory Fluc. Noise Lett. 1 R65-79
35. Planat M 2000 1/f frequency noise in a communication receiver and the Riemann hypothesis Lect. Notes Phys. 550 265-87
36. Planat M and Baboin A C 2007 Qudits of composite dimension, mutually unbiased bases and projective ring geometry J. Phys. A: Math. Theor. 40 F1-8
37. Weigert S and Wilkinson M 2008 Mutually unbiased bases for continuous variables Preprint quant-ph/0802.0394
38. Rains E M, Hardin R H, Schor P W and Sloane N J A 1997 Phys. Rev. Lett. 79 953-54
39. Bombin H and Martin-Delgado M A 2007 A family of non-Abelian Kitaev models on a lattice: topological confinement and condensation Preprint cond-mat.str-el/0712.0190