Introduction to Grassmann manifolds and quantum computation

Journal of Applied Mathematics

Volumn 2, Issue 8, 2002, Pages 371-405

  Fujii, Kazuyuki ^a  

^a NONE

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0242292437     PISSN: 1110757X     EISSN: None     Source Type: Journal    
DOI: 10.1155/S1110757X02110163     Document Type: Review

References (31)

1. A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. H. Margolus, P. W. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter, Elementary gates for quantum computation, Phys. Rev. A 52 (1995), no. 5, 3457-3467.
2. G. Chen and Z. Diao, Exponentially fast quantum search algorithm, http://arxiv.org/abs/math/quant-ph/0011109.
3. D. Deutsch, A. Barenco, and A. Ekert, Universality in quantum computation, Proc. Roy. Soc. London Ser. A 449 (1995), no. 1937, 669-677.
4. G. Dunne, Self-Dual Chern-Simons Theories, Lecture Notes in Physics, vol. M36, Springer, Berlin, 1995.
5. K. Fujii, Mathematical foundations of holonomic quantum computer II, http://arxiv.org/abs/quant-ph/0101102
6. K. Fujii, More on optical holonomic quantum computer, http://arxiv.org/abs/math/quant-ph/0005129
7. K. Fujii, Note on coherent states and adiabatic connections, curvatures, J. Math. Phys. 41 (2000), no. 7, 4406-4412.
8. K. Fujii, Mathematical foundations of holonomic quantum computer, Rep. Math. Phys. 48 (2001), no. 1-2, 75-82.
9. K. Fujii, A lecture on quantum logic gates, The Bulletin of Yokohama City University 53 (2002), 1-10.
10. K. Fujii, T. Kashiwa, and S. Sakoda, Coherent states over Grassmann manifolds and the WKB exactness in path integral, J. Math. Phys. 37 (1996), no. 2, 567-602.
11. K. Funahashi, private communication.
12. L. K. Grover, A framework for fast quantum mechanical algorithms, STOC 98 (Dallas, TX), ACM, New York, 1999, pp. 53-62.
13. A. Hosoya, Lectures on Quantum Computation, Science Publishing, Peking, 1999.
14. R. Jozsa, Quantum algorithms and the Fourier transform , R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454 (1998), no. 1969, 323-337.
15. R. Jozsa, Quantum factoring, discrete logarithms and the hidden subgroup problem, IEEE Computing in Science and Engineering 3 (2001), no. 2, 34-43.
16. A. Yu. Kitaev, Fault-tolerant quantum computation by anyons, http://arxiv.org/abs/quant-ph/9707021
17. H.-K. Lo, S. Popescu, and T. Spiller (eds.), Introduction to Quantum Computation and Information, World Scientific Publishing, New Jersey, 1998.
18. M. Nakahara, Geometry, Topology and Physics, Graduate Student Series in Physics, Adam Hilger, Bristol, 1990.
19. H. Oike, Geometry of Grassmann Manifolds, Yamagata University, Japan, 1979.
20. J. Pachos and S. Chountasis, Optical holonomic quantum computer, Phys. Rev. A 62 (2000), 1-9, 052318.
21. J. Pachos and P. Zanardi, Quantum holonomies for quantum computing, Internat. J. Modern Phys. B 15 (2001), no. 9, 1257-1285.
22. J. Pachos, P. Zanardi, and M. Rasetti, Non-Abelian Berry connections for quantum computation, Phys. Rev. A (3) 61 (2000), no. 1, 1-4, 010305.
23. R. F. Picken, The Duistermaat-Heckman integration formula on flag manifolds, J. Math. Phys. 31 (1990), no. 3, 616-638.
24. J. Preskill, Fault-tolerant quantum computation, Introduction to Quantum Computation and Information (H.-K. Lo, S. Popescu, and T. Spiller, eds.), World Scientific Publishing, New Jersey, 1998, pp. 213-269.
25. S. G. Rajeev, S. K. Rama, and S. Sen, Symplectic manifolds, coherent states, and semiclassical approximation, J. Math. Phys. 35 (1994), no. 5, 2259-2269.
26. E. G. Rieffel and W. Polak, An introduction to quantum computing for non-physicists, ACM Comput. Surveys 32 (2000), no. 3, 300-335.
27. A. Shapere and F. Wilczek (eds.), Geometric Phases in Physics, Advanced Series in Mathematical Physics, vol. 5, World Scientific Publishing, New Jersey, 1989.
28. P. W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM J. Comput. 26 (1997), no. 5, 1484-1509.
29. A. Steane, Quantum computing, Rep. Progr. Phys. 61 (1998), no. 2, 1 17-173.
30. W. J. Zakrzewski, Low-Dimensional Sigma Models, Adam Hilger, Bristol, 1989.
31. P. Zanardi and M. Rasetti, Holonomic quantum computation, Phys. Lett. A 264 (1999), no. 2-3, 94-99.