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Quantum Speedup and Categorical Distributivity

  • Peter Hines
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7860)

Abstract

This paper studies one of the best-known quantum algorithms — Shor’s factorisation algorithm — via categorical distributivity. A key aim of the paper is to provide a minimal set of categorical requirements for key parts of the algorithm, in order to establish the most general setting in which the required operations may be performed efficiently.

We demonstrate that Laplaza’s theory of coherence for distributivity [13,14] provides a purely categorical proof of the operational equivalence of two quantum circuits, with the notable property that one is exponentially more efficient than the other. This equivalence also exists in a wide range of categories.

When applied to the category of finite-dimensional Hilbert spaces, we recover the usual efficient implementation of the quantum oracles at the heart of both Shor’s algorithm and quantum period-finding generally; however, it is also applicable in a much wider range of settings.

Keywords

Category Theory Quantum Computing Shor’s Algorithm Monoidal Tensors Distributivity Coherence 

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References

  1. 1.
    Abramsky, S., Coecke, B.: A categorical semantics of quantum protocols. In: Proc. 19th Annual IEEE Symp. on Logic in Computer Science (LICS 2004), pp. 415–425. IEEE Computer Soc. Press (2005)Google Scholar
  2. 2.
    Abramsky, S.: Abstract Scalars, Loops, and Free Traced and Strongly Compact Closed Categories. In: Fiadeiro, J.L., Harman, N.A., Roggenbach, M., Rutten, J. (eds.) CALCO 2005. LNCS, vol. 3629, pp. 1–29. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  3. 3.
    Aharanov, D., Landau, Z., Makowsky, J.: The quantum FFT can be classically simulated, arXiv:quant-ph/0611156 v1 (2006)Google Scholar
  4. 4.
    Barr, M.: Algebraically Compact Functors. Journal of Pure and Applied Algebra 82, 211–231 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Blute, R.F., Cockett, J.R.B., Seely, R.A.G., Trimble, T.H.: Natural deduction and coherence for weakly distributive categories. Mathematical Structures in Computer Science 113, 229–296 (1991)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Carboni, A., Lack, S., Walters, R.: Introduction to Extensive and Distributive Categories. Journal of Pure and Applied Algebra 84, 145–158 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cockett, J.R.B.: Introduction to Distributive Categories. Mathematical Structures in Computer Science 3, 277–307 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Coecke, B., Pavlovic, D.: Quantum measurements without sums. In: Chen, G., Kauffman, L., Lamonaco, S. (eds.) Mathematics of Quantum Computing and Technology. Taylor and Francis (arxiv.org/quant-ph/0608035) (2007)Google Scholar
  9. 9.
    Hines, P.: Quantum circuit oracles for abstract machine computations. Theoretical Computer Science 411, 1501–1520 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Høyer, P., Špalek, R.: Quantum Fan-out is Powerful. Theory of Computing 1(5), 81–103 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Joyal, A., Street, R.: The geometry of tensor calculus. Advances in Mathematics (102), 20–78 (1993)Google Scholar
  12. 12.
    Joyal, A., Street, R.: The geometry of tensor calculus II (manuscript)Google Scholar
  13. 13.
    Laplaza, M.: Coherence for categories with associativity, commutativity, and distributivity. Bulletin of the American Mathematical Society 72(2), 220–222 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Laplaza, M.: Coherence for distributivity. In: MacLane, S. (ed.) Coherence in Categories. Springer Lecture Notes in Mathematics, vol. 281, pp. 29–65 (1972)Google Scholar
  15. 15.
    MacLane, S.: Duality for groups. Bulletin of the American Mathematical Society 56(6), 485–516 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    MacLane, S.: Categories for the working mathematician, 2nd edn. Springer, New York (1998)zbMATHGoogle Scholar
  17. 17.
    Nielsen, M., Chuang, I.: Quantum Computation and Quantum Information. Cambridge University Press (2000)Google Scholar
  18. 18.
    Pati, A., Braunstein, S.: Impossibility of deleting an unknown quantum state. Nature 404, 164–165 (2000)Google Scholar
  19. 19.
    Shor, P.: Algorithms for quantum computation: discrete log and factoring. In: Proceedings of IEEE FOCS, pp. 124–134 (1994)Google Scholar
  20. 20.
    Wootters, W., Zurek, W.: A Single Quantum Cannot be Cloned. Nature 299, 802–803 (1982)CrossRefGoogle Scholar
  21. 21.
    Yoran, N., Short, A.: Classical simulability and the significance of modular exponentiation in Shor’s algorithm, arXiv:quant-ph/0706.0872 v1 (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Peter Hines
    • 1
  1. 1.University of YorkUK

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