Skip to main content
SpringerLink
Log in

Quantum Information

  • Chapter
  • First Online:
Book cover Supervised Learning with Quantum Computers

Part of the book series: Quantum Science and Technology ((QST))

Abstract

While the last chapter introduced readers without a firm background in machine learning into the foundations, this chapter targets readers who are not closely familiar with quantum information. Quantum theory is notorious for being complicated, puzzling and difficult (or even impossible) to understand. Although this impression is debatable, giving a short introduction into quantum theory is indeed a challenge for two reasons: On the one hand its backbone, the mathematical apparatus with its objects and equations requires some solid mathematical foundations in linear algebra, and is usually formulated in the Dirac notation [1] that takes a little practice to get familiar with.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Common didactic approaches are the historical account of discovering quantum theory, the empirical account of experiments and their explanations, the Hamiltonian path from formal classical to quantum mechanics, the optical approach of the wave-particle-dualism, and the axiomatic postulation of its mathematical structure [2].

  2. 2.

    The lecture notes are available at http://www.scottaaronson.com/democritus/, and led to the book “Quantum Computing since Democritus” [3].

  3. 3.

    It shall be remarked that quantum theory has no notion of the concept of gravity, an open problem troubling physicists who dream of the so called ‘Grand Unified Theory’ of quantum mechanics and general relativity, and a hint towards the fact that quantum mechanics still has to be developed further.

  4. 4.

    In this sense, Aaronson is right in saying that the “central conceptual point” of quantum theory is “that nature is described not by probabilities [...], but by numbers called amplitudes that can be positive, negative, or even complex”.

  5. 5.

    The spectral theorem of linear algebra guarantees that the eigenvectors of a Hermitian operator O form a basis of the Hilbert space, so that every amplitude vector can be decomposed into eigenvectors of O.

  6. 6.

    From this form of an operator one can see that the density matrix is also an operator on Hilbert space, although it describes a quantum state.

  7. 7.

    This formalism is also known as the Lüders postulate.

  8. 8.

    http://math.nist.gov/quantum/zoo/.

  9. 9.

    Adapted from https://tex.stackexchange.com/questions/345420/how-to-draw-a-bloch-sphere.

  10. 10.

    http://www.dwavesys.com/.

  11. 11.

    The quadratic speed-up seems to be intrinsic in the structure of quantum probabilities, and derives from the fact that one has to square amplitudes in order to get probabilities.

References

  1. Dirac, P.A.M.: The Principles of Quantum Mechanics. Clarendon Press (1958)

    Google Scholar 

  2. Gerthsen, K., Vogel, H.: Physik. Springer (2013)

    Google Scholar 

  3. Aaronson, S.: Quantum Computing Since Democritus. Cambridge University Press (2013)

    Google Scholar 

  4. Leifer, M.S., Poulin, D.: Quantum graphical models and belief propagation. Ann. Phys. 323(8), 1899–1946 (2008)

    Google Scholar 

  5. Landau, L., Lifshitz, E.: Quantum mechanics: non-relativistic theory. In: Course of Theoretical Physics, vol. 3. Butterworth-Heinemann (2005)

    Google Scholar 

  6. Whitaker, A.: Einstein, Bohr and the Quantum Dilemma: From Quantum Theory to Quantum Information. Cambridge University Press (2006)

    Google Scholar 

  7. Einstein, A.: Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Annalen der Physik 322(8), 549–560 (1905)

    Article  ADS  Google Scholar 

  8. Einstein, A.: Zur Elektrodynamik bewegter Körper. Annalen der Physik 322(10), 891–921 (1905)

    Article  ADS  Google Scholar 

  9. Wilce, A.: Quantum logic and probability theory. In: Zalta, E.N. (ed.), The Stanford Encyclopedia of Philosophy. Online encyclopedia (2012). http://plato.stanford.edu/archives/fall2012/entries/qt-quantlog/

  10. Rédei, M., Summers, S.J.: Quantum probability theory. Studies in Hist. Phil. Sci. Part B Stud. Hist. Phil. Modern Phys. 38(2), 390–417 (2007)

    Google Scholar 

  11. Breuer, H.-P., Petruccione, F.: The Theory of Open Quantum Systems. Oxford University Press (2002)

    Google Scholar 

  12. Denil, M., De Freitas, N.: Toward the implementation of a quantum RBM. In: NIPS 2011 Deep Learning and Unsupervised Feature Learning Workshop (2011)

    Google Scholar 

  13. Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, pp. 212–219. ACM (1996)

    Google Scholar 

  14. Deutsch, D., Jozsa, R.: Rapid solution of problems by quantum computation. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 439, pp. 553–558. The Royal Society (1992)

    Google Scholar 

  15. Peter, W.: Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26(5), 1484–1509 (1997)

    Article  MathSciNet  Google Scholar 

  16. Deutsch, D.: Quantum theory, the Church-Turing principle and the universal quantum computer. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 400, pp. 97–117. The Royal Society (1985)

    Google Scholar 

  17. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2010)

    Google Scholar 

  18. Barenco, A., Bennett, C.H., Cleve, R., DiVincenzo, D.P., Margolus, N., Shor, P., Sleator, T., Smolin, J.A., Weinfurter, H.: Elementary gates for quantum computation. Phys. Rev. A, 52(5), 34–57 (1995)

    Google Scholar 

  19. Clarke, J., Wilhelm, F.K.: Superconducting quantum bits. Nature 453(7198), 1031–1042 (2008)

    Google Scholar 

  20. Kok, P., Munro, W.J., Nemoto, K., Ralph, T.C., Dowling, J.P., Milburn, G.J.: Linear optical quantum computing with photonic qubits. Rev. Modern Phys. 79(1), 135 (2007)

    Google Scholar 

  21. Juan, I.: Cirac and Peter Zoller. Quantum computations with cold trapped ions. Phys. Rev. Lett. 74(20), 4091 (1995)

    Article  Google Scholar 

  22. Nayak, C., Simon, S.H., Stern, A., Freedman, M., Sarma, S.D.: Non-Abelian anyons and topological quantum computation. Rev. Modern Phys. 80(3), 1083 (2008)

    Google Scholar 

  23. Farhi, E., Goldstone, J., Gutmann, S., Sipser, M.: Quantum computation by adiabatic evolution (2000). arXiv:quant-ph/0001106, MIT-CTP-2936

  24. Das, A., Chakrabarti, B.K.: Colloquium: quantum annealing and analog quantum computation. Rev. Modern Phys. 80(3), 1061 (2008)

    Google Scholar 

  25. Neven, H., Denchev, V.S., Rose, G., Macready, W.G.: Training a large scale classifier with the quantum adiabatic algorithm. arXiv:0912.0779 (2009)

  26. Heim, B., Rønnow, T.F., Isakov, S.V., Troyer, M.: Quantum versus classical annealing of Ising spin glasses. Science 348(6231), 215–217 (2015)

    Google Scholar 

  27. Santoro, G.E., Tosatti, E.: Optimization using quantum mechanics: quantum annealing through adiabatic evolution. J. Phys. A 39(36), R393 (2006)

    Google Scholar 

  28. Briegel, H.J., Raussendorf, R.: Persistent entanglement in arrays of interacting particles. Phys. Rev. Lett. 86(5), 910 (2001)

    Google Scholar 

  29. Nielsen, M.A.: Cluster-state quantum computation. Reports Math. Phys. 57(1), 147–161 (2006)

    Google Scholar 

  30. Prevedel, R., Stefanov, A., Walther, P., Zeilinger, A.: Experimental realization of a quantum game on a one-way quantum computer. New J. Phys. 9(6), 205 (2007)

    Google Scholar 

  31. Park, H.S., Cho, J., Kang, Y., Lee, J.Y., Kim, H., Lee, D.-H., Choi, S.-K.: Experimental realization of a four-photon seven-qubit graph state for one-way quantum computation. Opt. Exp. 20(7), 6915–6926 (2012)

    Google Scholar 

  32. Tame, M.S., Prevedel, R., Paternostro, M., Böhi, P., Kim, M.S., Zeilinger, A.: Experimental realization of Deutsch’s algorithm in a one-way quantum computer. Phys. Rev. Lett. 98, 140–501 (2007)

    Article  MathSciNet  Google Scholar 

  33. Tame, M.S., Bell, B.A., Di Franco, C., Wadsworth, W.J., Rarity, J.G.: Experimental realization of a one-way quantum computer algorithm solving Simon’s problem. Phys. Rev. Lett. 113, 200–501 (2014)

    Article  Google Scholar 

  34. Lloyd, S., Braunstein, S.L.: Quantum computation over continuous variables. In: Quantum Information with Continuous Variables, pp. 9–17. Springer (1999)

    Google Scholar 

  35. Weedbrook, C., Pirandola, S., García-Patrón, R., Cerf, N.J., Ralph, T.C., Shapiro, J.H., Lloyd, S.: Gaussian quantum information. Rev. Modern Phys. 84(2), 621 (2012)

    Google Scholar 

  36. Lau, H.-K., Pooser, R., Siopsis, G., Weedbrook, C.: Quantum machine learning over infinite dimensions. Phys. Rev. Lett. 118(8), 080501 (2017)

    Google Scholar 

  37. Chatterjee, R., Ting, Y.: Generalized coherent states, reproducing kernels, and quantum support vector machines. Quant. Inf. Commun. 17(15, 16), 1292 (2017)

    Google Scholar 

  38. Schuld, M., Killoran, N.: Quantum machine learning in feature Hilbert spaces. arXiv:1803.07128v1 (2018)

  39. Gisin, N.: Weinberg’s non-linear quantum mechanics and supraluminal communications. Phys. Lett. A 143(1), 1–2 (1990)

    Google Scholar 

  40. Polchinski, J.: Weinberg’s nonlinear quantum mechanics and the Einstein-Podolsky-Rosen paradox. Phys. Rev. Lett. 66, 397–400 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  41. Daniel, S.: Abrams and Seth Lloyd. Nonlinear quantum mechanics implies polynomial-time solution for NP-complete and #P problems. Phys. Rev. Lett. 81(18), 39–92 (1998)

    Google Scholar 

  42. Peres, A.: Nonlinear variants of Schrödinger’s equation violate the second law of thermodynamics. Phys. Rev. Lett. 63(10), 1114 (1989)

    Article  ADS  Google Scholar 

  43. Yoder, T.L., Low, G.H., Chuang, I.L.: Quantum inference on Bayesian networks. Phys. Rev. A 89, 062315 (2014)

    Google Scholar 

  44. Ozols, M., Roetteler, M., Roland, J.: Quantum rejection sampling. ACM Trans. Comput. Theory (TOCT) 5(3), 11 (2013)

    Google Scholar 

  45. Aram, W., Harrow, A.H., Lloyd, S.: Quantum algorithm for linear systems of equations. Phys. Rev. Lett. 103(15), 150–502 (2009)

    Google Scholar 

  46. Boyer, M., Brassard, G., Høyer, P., Tapp, A.: Tight bounds on quantum searching. Fortsch. Phys. 46, 493–506 (1998)

    Google Scholar 

  47. Ambainis, A.: Quantum lower bounds by quantum arguments. In: Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, pp. 636–643. ACM (2000)

    Google Scholar 

  48. Biham, E., Biham, O., Biron, D., Grassl, M., Lidar, D.A.: Grovers quantum search algorithm for an arbitrary initial amplitude distribution. Phys. Rev. A 60(4), 27–42 (1999)

    Google Scholar 

  49. Brassard, G., Høyer, P., Mosca, M., Tapp, A.: Quantum amplitude amplification and estimation. Contemp. Math. 305, 53–74 (2000)

    Google Scholar 

  50. Watrous, J.: Theory of Quantum Information. Cambridge University Press (2018)

    Google Scholar 

  51. Abrams, D.S., Lloyd, S.: Quantum algorithm providing exponential speed increase for finding eigenvalues and eigenvectors. Phys. Rev. Lett. 83(24), 51–62 (1999)

    Google Scholar 

  52. Clader, D.B., Jacobs, B.C., Sprouse, C.R.: Preconditioned quantum linear system algorithm. Phys. Rev. Lett. 110(25), 250–504 (2013)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Chemistry and Physics, Quantum Research Group, University of KwaZulu-Natal, Durban, South Africa

    Maria Schuld

  2. National Institute for Theoretical Physics (NITheP), KwaZulu-Natal, South Africa

    Maria Schuld & Francesco Petruccione

  3. Xanadu Quantum Computing Inc, Toronto, Canada

    Maria Schuld

  4. School of Chemistry and Physics, University of KwaZulu-Natal, Durban, South Africa

    Francesco Petruccione

  5. School of Electrical Engineering, Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Republic of Korea

    Francesco Petruccione

Authors
  1. Maria Schuld
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Francesco Petruccione
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maria Schuld .

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Schuld, M., Petruccione, F. (2018). Quantum Information. In: Supervised Learning with Quantum Computers. Quantum Science and Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-96424-9_3

Download citation

Publish with us

Policies and ethics

Search

Navigation