Hamming Distance Kernelisation via Topological Quantum Computation

  • Alessandra Di Pierro
  • Riccardo Mengoni
  • Rajagopal Nagarajan
  • David Windridge
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10687)

Abstract

We present a novel approach to computing Hamming distance and its kernelisation within Topological Quantum Computation. This approach is based on an encoding of two binary strings into a topological Hilbert space, whose inner product yields a natural Hamming distance kernel on the two strings. Kernelisation forges a link with the field of Machine Learning, particularly in relation to binary classifiers such as the Support Vector Machine (SVM). This makes our approach of potential interest to the quantum machine learning community.

Keywords

Quantum computing Topology Kernel function 

References

  1. 1.
    Adams, C.: The Knot Book. W.H. Freeman, New York (1994)MATHGoogle Scholar
  2. 2.
    Aharonov, D., Jones, V., Landau, Z.: A polynomial quantum algorithm for approximating the Jones polynomial. In: Proceedings of the 38th Annual ACM Symposium on Theory of Computing, Seattle, WA, USA, 21–23 May 2006, pp. 427–436 (2006)Google Scholar
  3. 3.
    Alexander, J.W.: A lemma on systems of knotted curves. Proc. Natl. Acad. Sci. U.S.A. 9(3), 93–95 (1923)CrossRefGoogle Scholar
  4. 4.
    Markoff, A.: Uber die freie äquivalenz der geschlossenen zöpfe. Rec. Math. [Mat. Sbornik] N.S. (1936)Google Scholar
  5. 5.
    Cortes, C., Vapnik, V.: Support-vector networks. Mach. Learn. 20(3), 273–297 (1995)MATHGoogle Scholar
  6. 6.
    Freedman, M.H.: P/NP, and the quantum field computer. Proc. Natl. Acad. Sci. 95(1), 98–101 (1998)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Freedman, M.H., Kitaev, A., Wang, Z.: Simulation of topological field theories by quantum computers. Commun. Math. Phys. 227, 587–603 (2002)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hamming, R.W.: Error detecting and error correcting codes. Bell System Tech J. 29, 147–160 (1950)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Jones, V.F.R.: A polynomial invariant for knots via von Neumann algebras. Bull. Amer. Math. Soc. (N.S.) 12(1), 103–111 (1985)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Kauffman, L.H.: State models and the Jones polynomial. Topology 26(3), 395–407 (1987)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Kauffman, L.H.: New invariants in the theory of knots. Am. Math. Monthly 95(3), 195–242 (1988)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Kauffman, L.H.: Knots and Physics. Series on Knots and Everything, 4th edn. World Scientific, Singapore (2013)CrossRefMATHGoogle Scholar
  13. 13.
    Kitaev, A., Preskill, J.: Topological entanglement entropy. Phys. Rev. Lett. 96, 110404 (2006)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kitaev, A.: Fault-tolerant quantum computation by anyons. Ann. Phys. 303(1), 2–30 (2003)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Pachos, J.K.: Introduction to Topological Quantum Computation. Cambridge University Press, New York (2012)CrossRefMATHGoogle Scholar
  16. 16.
    Reidemeister, K.: Knoten und Gruppen. Springer, Heidelberg (1932).  https://doi.org/10.1007/978-3-642-65616-3 CrossRefMATHGoogle Scholar
  17. 17.
    Reidemeister, K.: Elementare begründung der knotentheorie. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 5(1), 24–32 (1927)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Satō, H.: Algebraic Topology: An Intuitive Approach. Iwanami Series in Modern Mathematics. American Mathematical Society, Providence (1999)MATHGoogle Scholar
  19. 19.
    Wilczek, F.: Quantum mechanics of fractional-spin particles. Phys. Rev. Lett. 49, 957–959 (1982)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Alessandra Di Pierro
    • 1
  • Riccardo Mengoni
    • 1
  • Rajagopal Nagarajan
    • 2
  • David Windridge
    • 2
  1. 1.Dipartimento di InformaticaUniversità di VeronaVeronaItaly
  2. 2.Department of Computer ScienceMiddlesex UniversityLondonUK

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