Classical model for bulk-ensemble NMR quantum computation

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 60, Issue 6, 1999, Pages 4354-4362

  Schack, Rüdiger ^a,b   Caves, Carlton M ^a  

^a UNIVERSITY OF NEW MEXICO   (United States)

^b UNIVERSITY OF LONDON   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001618346     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.60.4354     Document Type: Article

References (40)

1. D. G. Cory, A. F. Fahmy, and T. F. Havel, Proc. Natl. Acad. Sci. USA 94, 1634 (1997).
2. N. A. Gershenfeld and I. L. Chuang, Science 275, 350 (1997).
3. W. S. Warren, Science 277, 1688 (1997).
4. M. A. Nielsen, E. Knill, and R. Laflamme, Nature (London) 396, 52 (1998).
5. D. G. Cory, M. D. Price, W. Maas, E. Knill, R. Laflamme, W. H. Zurek, T. F. Havel, and S. S. Somaroo, Phys. Rev. Lett. 81, 2152 (1998).
6. R. Laflamme, E. Knill, W. H. Zurek, P. Catasti, and S. V. S. Mariappan, Philos. Trans. R. Soc. London, Ser. A 356, 1941 (1998).
7. N. Linden, H. Barjat, and R. Freeman, Chem. Phys. Lett. 296, 61 (1998).
8. D. G. Cory, M. D. Price, and T. F. Havel, Physica D 120, 82 (1998).
9. J. A. Jones and M. Mosca, Phys. Rev. Lett. 83, 1050 (1999).
10. I. L. Chuang, N. Gershenfeld, and M. G. Kubinec, Phys. Rev. Lett. 80, 3408 (1998).
11. I. L. Chuang, N. Gershenfeld, M. G. Kubinec, and D. W. Leung, Proc. R. Soc. London, Ser. A 454, 447 (1998).
12. I. L. Chuang, L. M. K. Vandersypen, X. Zhou, D. W. Leung, and S. Lloyd, Nature (London) 393, 143 (1998).
13. E. Knill, I. Chuang, and R. Laflamme, Phys. Rev. A 57, 3348 (1998).
14. D. G. Cory, A. E. Dunlop, T. F. Havel, S. S. Somaroo, and W. Zhang, e-print quant-ph/9809045.
15. J. A. Jones, M. Mosca, and R. H. Hansen, Nature (London) 393, 344 (1998).
16. J. A. Jones and M. Mosca, J. Chem. Phys. 109, 1648 (1998).
17. J. A. Jones, R. H. Hansen, and M. Mosca, J. Magn. Reson., Ser. A 135, 353 (1998).
18. G. P. Berman, G. D. Doolen, G. V. López, and V. I. Tsifrinovich, Phys. Rev. B 58, 11 570 (1998).
19. E. Knill and R. Laflamme, Phys. Rev. Lett. 81, 5672 (1998).
20. L. J. Schulman and U. Vazirani, e-print quant-ph/9804060.
21. B. E. Kane, Nature (London) 393, 133 (1998).
22. D. P. DiVincenzo, review of Ref. 34 in Quick Reviews in Quantum Computation and Information, http://quickreviews.org
23. S. L. Braunstein, C. M. Caves, R. Jozsa, N. Linden, S. Popescu, and R. Schack, Phys. Rev. Lett. 83, 1054 (1999).
24. That entanglement is the “essential non-classical feature providing the computational speed-up in the known quantum algorithms” is argued forcefully by A. Ekert and R. Jozsa, Philos. Trans. R. Soc. London, Ser. A 356, 1769 (1998).
25. W. H. Zurek, in replying 6 to a question about whether several image-processing techniques can be regarded as quantum-information processing, expresses the following general view: “None of the examples invoked make an explicit use of arbitrary quantum superpositions or entanglement for information processing. Thus, while of interest, they cannot be regarded as ‘quantum computation’ (Formula presented).”
26. Knill and Laflamme 19 analyze the “power of one qubit of quantum information” by considering a model for ensemble computation using input states in which one qubit is in a pure state and all the other qubits are in a maximally mixed state. They note that although the apparent power of quantum computers “is frequently attributed to ‘quantum parallelism,’ interference phenomena derived from the superposition principle, and the ability to prepare and control pure states according to the Schrödinger equation,” their results suggest that “the usual reasons given for why quantum computation appears to be so powerful may have to be revised.”
27. M. A. Nielsen, Ph.D. dissertation, University of New Mexico, 1998. After analyzing a simple example related to Knill and Laflamme’s 19 “power of one qubit,” Nielsen speculates (p. 161), “(Formula presented) this example does suggest that the statement ‘entanglement is responsible for the power of quantum computation’ needs to be explored in much greater depth if it is to be made into a precise statement about the difference in computational power between quantum and classical computation.”
28. S. Lloyd, e-print quant-ph/9903057.
29. It is likely that there are Bohm-type “nonlocal” hidden-variable models for N qubits. Such a model would look like the Bohmian description of translational degrees of freedom [for an introduction to Bohmian mechanics, see the articles in Bohmian Mechanics and Quantum Theory: An Appraisal, edited by J. T. Cushing, A. Fine, and S. Goldstein (Kluwer, Dordrecht, 1996)] and would reproduce all the predictions of quantum mechanics by incorporating the state vector directly into the model. It would, however, violate the assumptions of our third criterion, in that the transition probabilities for a gate would depend on the initial state or would include qubits not involved in a gate in a nontrivial way.
30. Atta-ur-Rahman and M. I. Choudhary, Solving Problems with NMR Spectroscopy (Academic Press, San Diego, 1996).
31. G. Vidal and R. Tarrach, Phys. Rev. A 59, 141 (1999).
32. F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, Phys. Rev. A 6, 2211 (1972).
33. M. O. Scully and K. Wódkiewicz, Found. Phys. 24, 85 (1984).
34. K. Życzkowski, P. Horodecki, A. Sanpera, and M. Lewenstein, Phys. Rev. A 58, 883 (1998).
35. For a complete discussion, see Ref. 31 and references cited therein on measures of entanglement.
36. A. Peres, Quantum Theory: Concepts and Methods (Kluwer, Dordrecht, 1993).
37. Consider the case of two qubits (Formula presented), initially in a computational-basis state (Formula presented), which undergo a nonfactorizable unitary transformation U that takes computational-basis states to (entangled) Bell states. After the unitary dynamics, the two qubits are measured in the computational basis; let (Formula presented) denote the POVM elements of this measurement. The measurement statistics can be regarded as coming from a measurement of factorizable POVM (Formula presented) on the entangled state (Formula presented) or as a measurement of a nonfactorizable POVM (Formula presented) on the separable state (Formula presented). According to the discussion surrounding Eq. (3.9), the latter can be given a classical interpretation, whereas the former cannot. The discrepancy is resolved in the following way: introduce a tensor-product decomposition of the four-dimensional Hilbert space in which the computational-basis states and the Bell states switch roles, the original computational-basis states becoming Bell states and the original Bell states becoming product states (this new decomposition corresponds to a different way of dividing the two-qubit system into two subsystems); analyzed in this new tensor-product decomposition, the measurement of (Formula presented) on (Formula presented) becomes identical to the measurement of (Formula presented) on (Formula presented) and cannot be given a classical interpretation. These considerations are relevant to states of the form (2.1), with (Formula presented), for these states are separable relative to all decompositions of the (Formula presented)-dimensional Hilbert space into a tensor product of N two-dimensional spaces. This superseparability does not hold generally for separable states.
38. L. J. Landau and R. F. Streater, Linear Algebr. Appl. 193, 107 (1993).
39. Although Ref. 20 shows that bulk-ensemble NMR is scalable to many qubits, the proper scaling sets in at such a large number of qubits, (Formula presented), that it is irrelevant to the present paper.
40. Some authors have been careful about advancing claims for the quantum nature of NMR quantum-information processing. For example, Cory et al., in a paper 8 outlining the principles of NMR quantum computation, comment (in the abstract) that “nuclear magnetic resonance spectroscopy is capable of emulating (our emphasis) many of the capabilities of quantum computers, including unitary evolution and coherent superpositions, but without attendant wave-function collapse.” Others have been less cautious. Gershenfeld and Chuang, in their original paper on NMR 2, remark, “Because the N spins in each molecule may be in entangled quantum superposition states, our computer can be a quantum one,” and Chuang et al. 11, in describing the experiment referred to at the end of Sec. III, which implements a single two-qubit gate (a controlled-NOT), note, “Perhaps the most interesting aspect of this experiment is that the experimental results cannot be explained by a classical model of two interacting spins (of spin (Formula presented).”