Abstract
This paper provides necessary and sufficient conditions for constructing a universal quantum computer over continuous variables. As an example, it is shown how a universal quantum computer for the amplitudes of the electromagnetic field might be constructed using simple linear devices such as beam splitters and phase shifters, together with squeezers and nonlinear devices such as Kerr-effect fibers and atoms in optical cavities. Such a device could in principle perform “floating point” computations. Problems of noise, finite precision, and error correction are discussed.
S. Lloyd and S. L. Braunstein, Physical Review Letters 82, 1784–1787 (1999).
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References
D. DiVincenzo, Science 270, 255 (1995).
S. Lloyd, Sci. Am. 273, 140 (1995).
S. Lloyd and J. J.-E. Slotine, Phys. Rev. Lett. 80, 4088 (1998).
S. L. Braunstein, Phys. Rev. Lett. 80, 4084 (1998).
S. L. Braunstein, Nature (London) 394, 47 (1998).
S. L. Braunstein and H. J. Kimble, Phys. Rev. Lett. 80, 869 (1998).
A. Furusawa, et al, Science 282, 706 (1998).
This definition of quantum computation corresponds to the normal “circuit” definition of quantum computation as in, e.g., D. Deutsch, Proc. R. Soc. London A, 425, 73 (1989)
A. C.-C. Yao, in Proceedings of the 36th Annual Symposium on Foundations of Computer Science, edited by S. Goldwasser, (IEEE Computer Society, Los Alamitos, CA, 1995), pp. 352–361. The works of
M. Reck et al., Phys. Rev. Lett. 73, 58 (1994), and
N. J. Cerf, C. Adami, and P. G. Kwiat, Phys. Rev. A 57, R1477 (1998), showing how to perform arbitrary unitary operators using only linear devices such as beam splitters, though of considerable interest and potential practical importance, does not constitute quantum computation by the usual definition. Reck et al. and Cerf et al. propose performing arbitrary unitary operations on N variables not by acting on the variables themselves but by expanding the information in the variables into an interferometer with O(2N) arms and acting in this exponentially larger space. Local operations on the original variables correspond to highly nonlocal operations in this “unary” representation: To flip a single bit requires one to act on half [O(2N−1)] of the arms of the interferometer. Actually to perform quantum computation on qubits using an interferometer requires nonlinear operations as detailed in
N. Yang, K. Young (World Scientific, Singapore, 1988), pp. 779–799
G. J. Milburn, Phys. Rev. Lett. 62 2124 (1989).
G. M. Huang, T. J. Tarn, J. W. Clark, J. Math. Phys. (N.Y) 24, 2608–2618 (1983.
Differential Geometric Control Theory, edited by R. W. Brockett, R. S. Millman and H. J. Sussman, (Birkhauser, Boston, 1983); Nonholonomic Motion Planning, edited by Z. Li and J. F. Canney (Kluwer Academic, Boston, 1993).
V. Ramakrishna, M. V. Salapaka, M. Dahleh, H. Rabitz and A. Peirce, Phys. Rev. A 51, 960–966 (1995).
S. Lloyd, Phys. Rev. Lett. 75, 346–349 (1995).
D. Deutsch, A. Barenco and A. Ekert, Proc. R. Soc. London A 449, 669–677 (1995).
S. Lloyd, Science 273, 1073 (1996).
L. Blum, M. Shub and S. Smale, Bull. Am. Math. Soc. 21, 1–46 (1989).
L. A. Wu et al., Phys. Rev. Lett. 57, 2520 (1986).
Q. A. Turchette, et al, Phys. Rev. Lett. 75, 4710–4713 (1995).
R. Landauer, Nature (London) 335, 779–784 (1988).
R. Landauer, Phys. Lett. A 217, 188–193 (1996).
R. Landauer, Phil. Trans. R. Soc. London A 335, 367–376 (1995).
P. Shor, Proceedings of the 37th Annual Symposium on the Foundations of Computer Science (IEEE Computer Society Press, Los Alamitos, 1996), pp. 56–65.
D. P. DiVincenzo and P. W. Shor, Phys. Rev. Lett. 77, 3260–3263 (1996).
R. Laflamme, M. Knill and W.H. Zurek, Science 279, 342 (1998)
D. Aharanov and Ben-Or, quant-ph; J. Preskill, Proc. R. Soc. London A 454, 385 (1998).
C. H. Bennett and G. Brassard, in Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing, Bangalore, India, (IEEE Press, New York, 1984), pp. 175–179.
A. K. Ekert, et al, Phys. Rev. Lett. 69, 1293 (1992)
P. D. Townsend, J. G. Rarity and P. R. Tapster, Electronics Letters 29, 1291 (1993)
R. J. Hughes, et al, in Advances in Cryptology: Proceedings of Crypto 96, (Springer-Verlag, New York, 1997), pp. 329–343
A. Muller et al. Appl. Phys. Lett. 70, 793 (1997).
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Lloyd, S., Braunstein, S.L. (1999). Quantum Computation Over Continuous Variables. In: Braunstein, S.L., Pati, A.K. (eds) Quantum Information with Continuous Variables. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-1258-9_2
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DOI: https://doi.org/10.1007/978-94-015-1258-9_2
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