Liquid-state NMR simulations of quantum many-body problems

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 71, Issue 3, 2005, Pages

  Negrevergne, C ^a   Somma, R ^b,c   Ortiz, G ^b   Knill, E ^d   Laflamme, R ^a  

^a UNIVERSITY OF WATERLOO   (Canada)

^b LOS ALAMOS NATIONAL LABORATORY   (United States)

^c INSTITUTO BALSEIRO   (Argentina)

^d NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

POLYNOMIAL RESOURCES; QUANTUM ALGORITHMS; QUANTUM COMPUTERS; QUANTUM SPIN MODEL;

COMPUTER SIMULATION; MATHEMATICAL MODELS; MATHEMATICAL OPERATORS; MONTE CARLO METHODS; NUCLEAR MAGNETIC RESONANCE; POLYNOMIALS; QUANTUM CRYPTOGRAPHY;

QUANTUM THEORY;

EID: 18444397703     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.71.032344     Document Type: Article

References (41)

1. P. Shor, Proceedings of the 35th Annual IEEE Symposium on Foundations of Computer Science, FOCS'94, Santa Fe, New Mexico 1994, pp. 124-134.
2. L. K. Grover, Proceedings of the 28th Annual ACM Symposium on Theory of Computing (STOC), New York, 1996, pp. 212-219.
3. A. Ambainis, quant-ph/0311001.
4. A. Kitaev, quant-ph/9511026.
5. M. Mosca and A. Ekert, quant-ph/9903071.
6. R. P. Feynman, Int. J. Theor. Phys. 21, 467 (1982).
7. In Feynman's words, "If you had discrete quantum systems, what other discrete quantum systems are exact imitations of it, and is there a class against which everything can be matched? I believe it's rather simple to answer that question and to find the class, but I just haven't done it."
8. G. Ortiz, J. Gubernatis, E. Knill, and R. Laflamme, Phys. Rev. A 65, 042323 (2001).
9. C. D. Batista and G. Ortiz, Phys. Rev. Lett. 86, 1082 (2001).
10. R. Somma, G. Ortiz, J. E. Gubernatis, E. Knill, and R. Laflamme, Phys. Rev. A 65, 022319 (2002).
11. C. D. Batista and G. Ortiz, Adv. Phys. 53, 1 (2004).
12. S. Lloyd, Science 273, 1073 (1996).
13. For example, H. Q. Lin and J. E. Gubernatis, Comput. Phys. 7, 400 (1993).
14. D. M. Ceperley, Rev. Mod. Phys. 67, 279 (1995).
15. For example, B. B. Beard and U.-J. Wiese, Phys. Rev. Lett. 77, 5130 (1996).
16. R. Laflamme, E. Knill, D. G. Cory, E. M. Fortunate, T. Havel, C. Miquel, R. Martinez, C. Negrevergne, G. Ortiz, M. A. Pravia, Y. Sharf, S. Sinha, R. Somma, and L. Viola, Los Alamos Sci. 27, 226 (2002).
17. C. H. Tseng, S. Somaroo, Y. Sharf, E. Knill, R. Laflamme, T. F. Havel, and D. G. Cory, Phys. Rev. A 61, 012302 (2000).
18. A. K. Khitrin and B. M. Fung, Phys. Rev. A 64, 032306 (2001).
19. Experimental results were anticipated in the Lattice 2001 Conference in Berlin;
20. G. Ortiz, E. Knill, and J. E. Gubernatis, Nucl. Phys. B 106, 151 (2002). Here we present them in full detail.
21. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, UK, 2000).
22. A. Barenco et al., Phys. Rev. A 52, 3457 (1995).
23. D. DiVicenzo, Phys. Rev. A 51, 1015 (1995).
24. We actually restrict ourselves to 90° and 180° rotations for experimental calibration issues.
25. Note that, even though efficient techniques to prepare a pseudopure state exist in theory [25], in practice they are very hard to implement and one instead uses nonefficient methods that suffer a exponential decay of observed signal with respect to the number of qubits in the pseudopure state.
26. L. Schuman and U. Vazirani, Scalable NMR quantum computation., Proceedings of the 31st ACM Symposium on Theory of Computing, 1998, pp. 322-329.
27. P. Jordan and E. Wigner, Z. Phys. 47, 631 (1928).
28. R. Somma, G. Ortiz, E. Knill, and J. Gubernatis, Int. J. Quantum Inf. 1, 189 (2003).
29. i(π/4)σy.
30. E. Knill, R. Laflamme, R. Martinez, and C. Tseng, Nature (London) 404 368 (2000).
31. E. Knill, R. Laflamme, R. Martinez, and C. Negrevergne, Phys. Rev. Lett. 86, 5811 (2001).
32. By ideal we mean without experimental error and decoherence effects.
33. 1) on the other spins is a Larmor frequency shift that can be absorbed into the phase tracking calculation.
34. See, for example, P. W. Shor, Phys. Rev. A 52, R2493 (1995);
35. P. W. Shor, in Proceedings of the 37th Annual IEEE Symposium of Foundaments of Computer Science, FOC'96, Burlington, VT (1996), pp. 56-65.
36. A. M. Steane, Phys. Rev. Lett. 77, 793 (1996);
37. A. Y. Kitaev, Russ. Math. Surveys 52, 1191 (1997).
38. Using these techniques would require to achieve an error-rate per gate below a threshold value depending on the exact error correction method used. Although none of these correcting techniques have been used in the present work, quantum error correction codes have been successfully implemented on the same NMR system [30].
39. c,with p an integer number.
40. l) could be in principle different from zero. However, we consider only its real part because it contains all the desired information (e.g., eigenvalues).
41. J. Taylor, An Introduction to Error Analysis (University Science Books, Sausalito, CA, 1997).