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Hidden subspace algorithm in white noise analysis


Author: Jeremy J. Becnel
Journal: Trans. Amer. Math. Soc. 364 (2012), 5035-5055
MSC (2010): Primary 60H40; Secondary 81P68
DOI: https://doi.org/10.1090/S0002-9947-2012-05661-3
Published electronically: April 30, 2012
MathSciNet review: 2931321
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Abstract: In 2003, Professors Lomonaco and Kauffman developed an algorithm to find a hidden subspace of a functional. The algorithm was developed in the spirit of Feynman path integrals but lacked mathematical rigor. In this paper we make use of the framework of the White Noise Distribution Theory to supply the appropriate mathematical machinery to make the algorithm rigorous. In the process we construct a Gaussian measure for affine subspaces of a Hilbert space and develop a decomposition theorem for these measures.


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  • 1. Jeremy J. Becnel and Ambar N. Sengupta, An infinite-dimensional integral identity for the Segal-Bargmann transform, Proc. Amer. Math. Soc. 135 (2007), no. 9, 2995–3003. MR 2317978, https://doi.org/10.1090/S0002-9939-07-08995-2
  • 2. Jeremy J. Becnel, Delta function for an affine subspace, Taiwanese J. Math. 12 (2008), no. 9, 2269–2294. MR 2479055
  • 3. Jeremy J. Becnel and Ambar N. Sengupta, The Schwartz space: Tools for white noise analysis, preprint at http://www.math.lsu.edu/ preprint/, December 2004.
  • 4. I. M. Gel′fand and G. E. Shilov, Generalized functions. Vol. 1, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964 [1977]. Properties and operations; Translated from the Russian by Eugene Saletan. MR 0435831
    I. M. Gel′fand and G. E. Shilov, Generalized functions. Vol. 2, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1968 [1977]. Spaces of fundamental and generalized functions; Translated from the Russian by Morris D. Friedman, Amiel Feinstein and Christian P. Peltzer. MR 0435832
    I. M. Gel′fand and G. E. Shilov, Generalized functions. Vol. 3, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1967 [1977]. Theory of differential equations; Translated from the Russian by Meinhard E. Mayer. MR 0435833
    I. M. Gel′fand and N. Ya. Vilenkin, Generalized functions. Vol. 4, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964 [1977]. Applications of harmonic analysis; Translated from the Russian by Amiel Feinstein. MR 0435834
    I. M. Gel′fand, M. I. Graev, and N. Ya. Vilenkin, Generalized functions. Vol. 5, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1966 [1977]. Integral geometry and representation theory; Translated from the Russian by Eugene Saletan. MR 0435835
  • 5. Lars Hörmander, The analysis of linear partial differential operators. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR 717035
    Lars Hörmander, The analysis of linear partial differential operators. II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 257, Springer-Verlag, Berlin, 1983. Differential operators with constant coefficients. MR 705278
  • 6. Izumi Kubo and Yoshitaka Yokoi, A remark on the space of testing random variables in the white noise calculus, Nagoya Math. J. 115 (1989), 139–149. MR 1018088
  • 7. Hui-Hsiung Kuo, White noise distribution theory, Probability and Stochastics Series, CRC Press, Boca Raton, FL, 1996. MR 1387829
  • 8. Hui Hsiung Kuo, Gaussian measures in Banach spaces, Lecture Notes in Mathematics, Vol. 463, Springer-Verlag, Berlin-New York, 1975. MR 0461643
  • 9. Samuel J. Lomonaco and Louis H. Kauffman, Continuous quantum hidden subgroup algorithms, Proceedings of SPIE, vol. 5105, 2003, pp. 80-88.
  • 10. Nobuaki Obata, White noise calculus and Fock space, Lecture Notes in Mathematics, vol. 1577, Springer-Verlag, Berlin, 1994. MR 1301775

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Additional Information

Jeremy J. Becnel
Affiliation: Department of Mathematics and Statistics, Stephen F. Austin State University, Nacogdoches, Texas 75962-3040
Email: becneljj@sfasu.edu

DOI: https://doi.org/10.1090/S0002-9947-2012-05661-3
Keywords: White noise analysis, quantum computing, hidden subspace algorithm
Received by editor(s): July 21, 2010
Published electronically: April 30, 2012
Additional Notes: This research was supported by Stephen F. Austin Faculty Research Grant
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.