Exact and approximate unitary 2-designs and their application to fidelity estimation

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 80, Issue 1, 2009, Pages

  Dankert, Christoph ^a   Cleve, Richard ^a,b   Emerson, Joseph ^c   Livine, Etera ^b  

^a UNIVERSITY OF WATERLOO   (Canada)

^b PERIMETER INSTITUTE FOR THEORETICAL PHYSICS   (Canada)

^c UNIVERSITY OF WATERLOO   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

ANCILLA; CIRCUIT CONSTRUCTION; N QUBITS; QUANTUM CIRCUIT; QUANTUM INFORMATION; QUANTUM PROCESS; STOCHASTIC PROPERTIES; T-DESIGNS; UNITARY GROUP;

DESIGN;

QUANTUM OPTICS;

EID: 68549090875     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.80.012304     Document Type: Article

References (32)

1. A. Ambainis, M. Mosca, A. Tapp, and R. de Wolf, Proceedings of the 41st FOCS (IEEE Computer Society Press, Los Alamitos, CA, 2000), pp. 547-553.
2. D. DiVincenzo, D. Leung, and B. Terhal, IEEE Trans. Inf. Theory 48, 580 (2002). 10.1109/18.985948
3. J. Radishkran, M. Roetteler, and P. Sen, Lect. Notes Comput. Sci. 3580, 1399 (2005)
4. P. Sen, in Proceedings of the 21st Annual IEEE Conference of Computational Complexity (IEEE Computer Society Press, Los Alamitos, CA, 2006), pp. 274-287.
5. A. Ambainis and A. Smith, Lect. Notes Comput. Sci. 3122, 249 (2004).
6. P. Hayden, D. Leung, P. Shor, and A. Winter, Commun. Math. Phys. 250, 371 (2004) 10.1007/s00220-004-1087-6
7. C. H. Bennett, P. Hayden, D. Leung, P. Shor, and A. Winter, IEEE Trans. Inf. Theory 51, 56 (2005) 10.1109/TIT.2004.839476
8. A. Harrow, P. Hayden, and D. Leung, Phys. Rev. Lett. 92, 187901 (2004). 10.1103/PhysRevLett.92.187901
9. J. Emerson, Y. Weinstein, M. Saraceno, S. Lloyd, and D. Cory, Science 302, 2098 (2003). 10.1126/science.1090790
10. J. Emerson, R. Alicki, and K. Zyczkowski, J. Opt. B: Quantum Semiclassical Opt. 7, S347 (2005). 10.1088/1464-4266/7/10/021
11. A. Abeyesinghe, I. Devetak, P. Hayden, and A. Winter, e-print arXiv:quant-ph/0606225 Proc. R. Soc. A (to be published).
12. C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, Phys. Rev. A 54, 3824 (1996). 10.1103/PhysRevA.54.3824
13. W. Dür, M. Hein, J. I. Cirac, and H.-J. Briegel, Phys. Rev. A 72, 052326 (2005). 10.1103/PhysRevA.72.052326
14. R. Oliveira, O. Dahlsten, and M. Plenio, Phys. Rev. Lett. 98, 130502 (2007). 10.1103/PhysRevLett.98.130502
15. J. Renes, R. Blume-Kohout, A. Scott, and C. Caves, J. Math. Phys. 45, 2171 (2004) 10.1063/1.1737053
16. A. Klappenecker and M. Roetteler, in Proceedings of the 2005 IEEE International Symposium on Information Theory (IEEE, Adelaide, Australia, 2005), p. 1740-1744.
17. An equivalent definition of a unitary t -design is such that k=1 K DI (Uk) =0 for all nontrivial irreducible representations DI contained in the tensor power Vt V̄ t of the fundamental representation V and its conjugate. This characterization is especially relevant when analyzing the efficiency of "random circuit" constructions for generating pseudorandom sets of unitaries.
18. For example, t=1 corresponds to the conditions of a private quantum channel, and t=4 corresponds to the case of the state-distinction problem.
19. D. Gottesman, Ph.D. thesis, California Institue of Technology, 1997.
20. A. Yu. Kitaev, A. H. Shen, and M. N. Vyali, Classical and Quantum Computation (American Mathematical Society, Providence, RI, 2002), Vol. 47.
21. C. Dankert, MMath thesis, University of Waterloo, 2005.
22. M. A. Nielsen, Phys. Lett. A 303, 249 (2002). 10.1016/S0375-9601(02) 01272-0
23. B. Schumacher, Phys. Rev. A 54, 2614 (1996). 10.1103/PhysRevA.54.2614
24. N. Boulant, T. F. Havel, M. A. Pravia, and D. G. Cory, Phys. Rev. A 67, 042322 (2003). 10.1103/PhysRevA.67.042322
25. E. M. Fortunato, M. A. Pravia, N. Boulant, G. Teklemariam, T. F. Havel, and D. G. Cory, J. Chem. Phys. 116, 7599 (2002). 10.1063/1.1465412
26. Y. Weinstein, T. F. Havel, J. Emerson, N. Boulant, M. Saraceno, and D. Cory, J. Chem. Phys. 121, 6117 (2004). 10.1063/1.1785151
27. I. Chuang and M. Nielsen, J. Mod. Opt. 44, 2455 (1997)
28. J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. OBrien, M. A. Nielsen, and A. G. White, Phys. Rev. Lett. 90, 193601 (2003). 10.1103/PhysRevLett.90.193601
29. J. Emerson, Science 317, 1893 (2007). 10.1126/science.1145699
30. J. Emerson, E. Livine, and S. Lloyd, Phys. Rev. A 72, 060302 (R) (2005). 10.1103/PhysRevA.72.060302
31. P. Sen, Proceedings of the 21st Annual IEEE Conference on Computational Complexity (IEEE Computer Society Press, Los Alamitos, CA, 2006), pp. 274-287.
32. A. Ambainis and J. Emerson, Proceedings of the 22nd Annual IEEE Conference on Computational Complexity (IEEE Computer Society Press, Los Alamitos, CA, 2007), pp. 129-140.