Foundations of Physics

, Volume 49, Issue 10, pp 1200–1230 | Cite as

A Relational Time-Symmetric Framework for Analyzing the Quantum Computational Speedup

  • G. CastagnoliEmail author
  • E. Cohen
  • A. K. Ekert
  • A. C. Elitzur


The usual representation of quantum algorithms is limited to the process of solving the problem. We extend it to the process of setting the problem. Bob, the problem setter, selects a problem-setting by the initial measurement. Alice, the problem solver, unitarily computes the corresponding solution and reads it by the final measurement. This simple extension creates a new perspective from which to see the quantum algorithm. First, it highlights the relevance of time-symmetric quantum mechanics to quantum computation: the problem-setting and problem solution, in their quantum version, constitute pre- and post-selection, hence the process as a whole is bound to be affected by both boundary conditions. Second, it forces us to enter into relational quantum mechanics. There must be a representation of the quantum algorithm with respect to Bob, and another one with respect to Alice, from whom the outcome of the initial measurement, specifying the setting and thus the solution of the problem, must be concealed. Time-symmetrizing the quantum algorithm to take into account both boundary conditions leaves the representation to Bob unaltered. It shows that the representation to Alice is a sum over histories in each of which she remains shielded from the information coming to her from the initial measurement, not from that coming to her backwards in time from the final measurement. In retrospect, all is as if she knew in advance, before performing her problem-solving action, half of the information that specifies the solution of the problem she will read in the future and could use this information to reach the solution with fewer computation steps (oracle queries). This elucidates the quantum computational speedup in all the quantum algorithms examined.


Quantum computational speedup Temporal nolocality Retrocausality Temporal Bell inequalities Quantum algorithms Quantum computing Quantum information Relational quantum mechanics Time-symmetric quantum mechanics 



We wish to thank Yakir Aharonov and David Ritz Finkelstein for many helpful discussions.


  1. 1.
    Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proceedings of the 28th Annual ACM Symposium on the Theory of Computing, pp. 212–219. ACM Press, New York (1996)Google Scholar
  2. 2.
    Mosca, M., Ekert, A.K.: The hidden subgroup problem and Eigen value estimation on a Quantum Computer. Lecture Notes in Computer Science, vol. 1509 (1999)CrossRefGoogle Scholar
  3. 3.
    Ambainis, A.: Understanding quantum algorithms via query complexity. arXiv:1712.06349 (2017)
  4. 4.
    Jozsa, R.: Entanglement and quantum computation. Geometric issues in the foundations of science, Oxford University Press. arXiv:quant-ph/9707034 (1997)
  5. 5.
    Ekert, A.K., Jozsa, R.: Quantum algorithms: Entanglement enhanced information processing. arXiv:quant-ph/9803072 (1998)
  6. 6.
    Vedral, V.: The elusive source of quantum effectiveness. Found. Phys. 40(8), 1141–1154 (2010)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Aaronson, S., Ambainis, A.: Forrelation: a problem that optimally separates quantum from classical computing. arXiv:1411.5729 (2014)
  8. 8.
    Castagnoli, G., Finkelstein, D.R.: Theory of the quantum speedup. Proc. R. Soc. A 1799(457), 1799–1807 (2001)ADSCrossRefGoogle Scholar
  9. 9.
    Castagnoli, G.: The quantum correlation between the selection of the problem and that of the solution sheds light on the mechanism of the quantum speed up. Phys. Rev. A 82, 052334 (2010)ADSCrossRefGoogle Scholar
  10. 10.
    Castagnoli, G.: Completing the physical representation of quantum algorithms provides a quantitative explanation of their computational speedup. Found. Phys. 48, 333–354 (2018)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Von Neumann, J.: Mathematical foundations of quantum mechanics. Princeton University Press, Princeton (1955)zbMATHGoogle Scholar
  12. 12.
    De Beauregard, C.O.: The 1927 Einstein and 1935 EPR paradox. Phys. A 2, 211–242 (1980)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Dolev, S., Elitzur, A.C.: Non-sequential behavior of the wave function. arXiv:quant-ph/0102109 v1 (2001)
  14. 14.
    Elitzur, A.C., Cohen, E.: Quantum oblivion: a master key for many quantum riddles. Int. J. Quant. Inf. 12, 1560024 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Elitzur, A.C., Cohen, E.: 1-1 = Counterfactual: on the potency and significance of quantum non-events. Phil. Trans. R. Soc. A 374, 20150242 (2016)ADSCrossRefGoogle Scholar
  16. 16.
    Wheeler, J.A., Feynman, R.P.: Interaction with the absorber as the mechanism of radiation. Rev. Mod. Phys. 17, 157–181 (1945)ADSCrossRefGoogle Scholar
  17. 17.
    Watanabe, S.: Symmetry of physical laws. Part III. Prediction and retrodiction. Rev. Mod. Phys. 27(2), 179–186 (1955)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Aharonov, Y., Bergman, P.G., Lebowitz, J.L.: Time symmetry in the quantum process of measurement. Phys. Rev. 134, 1410–1416 (1964)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Cramer, J.: The transactional interpretation of quantum mechanics. Rev. Mod. Phys. 58, 647 (1986)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Aharonov, Y., Rohrlich, D.: Quantum Paradoxes. Wiley-VCH, Weinheim (2005)CrossRefGoogle Scholar
  21. 21.
    Aharonov, Y., Vaidman, L.: The two-state vector formalism: an updated review. Lect. Notes Phys. 734, 399–447 (2008)ADSCrossRefGoogle Scholar
  22. 22.
    Aharonov, Y., Colombo, F., Popescu, S., Sabadini, I., Struppa, D.C., Tollaksen, J.: Quantum violation of the pigeonhole principle and the nature of quantum correlations. Proc. Natl. Acad. Sci. USA 113, 532–535 (2016)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Aharonov, Y., Cohen, E., Landsberger, T.: The two-time interpretation and macroscopic time-reversibility. Entropy 19, 111 (2017)ADSCrossRefGoogle Scholar
  24. 24.
    Aharonov, Y., Cohen, E., Tollaksen, J.: Completely top-down hierarchical structure in quantum mechanics. Proc. Natl. Acad. Sci. USA 115, 11730–11735 (2018)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Aharonov, Y., Cohen, E., Carmi, A., Elitzur, A.C.: Extraordinary interactions between light and matter determined by anomalous weak values. Proc. R. Soc. A 474, 20180030 (2018)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Bennett, C.H., Bernstein, E., Brassard, G., Vazirani, U.: Strengths and weaknesses of quantum computing. SIAM J. Comput. 26, 1510–1523 (1997)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Long, G.L.: Grover algorithm with zero theoretical failure rate. Phys. Rev. A 64, 022307–022314 (2001)ADSCrossRefGoogle Scholar
  28. 28.
    Toyama, F.M., van Dijk, W., Nogami, Y.: Quantum search with certainty based on modified Grover algorithms: optimum choice of parameters. Quant. Inf. Proc. 12, 1897–1914 (2013)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Deutsch, D., Jozsa, R.: Rapid solution of problems by quantum computation. Proc. R. Soc. A 439, 553–558 (1992)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Simon, D.: On the power of quantum computation. In: Proceedings of the 35th annual IEEE symposium on the foundations of computer science, pp. 116–123 (1994)Google Scholar
  31. 31.
    Shor, P.: Algorithms for quantum computation: discrete log and factoring. In: Proceedings of the 35th annual IEEE symposium on the foundations of computer science, pp. 124–131 (1994)Google Scholar
  32. 32.
    Kaye, P., Laflamme, R., Mosca, M.: An Introduction to Quantum Computing, pp. 146–147. Oxford University Press, Oxford (2007)zbMATHGoogle Scholar
  33. 33.
    Rovelli, C.: Relational quantum mechanics. Int. J. Theor. Phys. 35, 637–658 (1996)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Rovelli, C.: Relational quantum mechanics. (2011)
  35. 35.
    Fuchs, C. A.: On participatory realism. arXiv:1601.04360v3 [quant-ph] (2016)Google Scholar
  36. 36.
    Fuchs, C. A.: QBism, the perimeter of quantum Bayesianism. arXiv:1003.5209v1 [quant-ph] (2010)
  37. 37.
    Healey, R.: Quantum theory: a pragmatist approach. arXiv:1008.3896 (2010)
  38. 38.
    Adlam, E.: Spooky action at a temporal distance. Entropy 20(1), 41 (2018)ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    Aharonov, Y., Cohen, E., Elitzur, A.C.: Can a future choice affect a past measurement outcome? Ann. Phys. 355, 258–268 (2015)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    Aharonov, Y., Cohen, E., Shushi, T.: Accommodating retrocausality with free will. Quanta 5, 53–60 (2016)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Carmi, A., Cohen, E.: Relativistic independence bounds nonlocality. Sci. Adv. 5, eaav8370 (2019)ADSCrossRefGoogle Scholar
  42. 42.
    Morikoshi, F.: Information-theoretic temporal Bell inequality and quantum computation. Phys. Rev. A 73, 052308 (2006)ADSCrossRefGoogle Scholar
  43. 43.
    Finkelstein, D.R.: Space-time structure in high energy interactions. In: Gudehus, T., Kaiser, G., Perlmutter, A. (eds.) Conference on high energy interactions, Coral Gables (1968)Google Scholar
  44. 44.
    Finkelstein, D.R.: Space-time code. Phys. Rev. 184, 1261 (1969)ADSMathSciNetCrossRefGoogle Scholar
  45. 45.
    Feynman, R.P.: Simulating physics with computers. Int. J. Theor. Phys. 21, 467–488 (1982)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Bennett, C.H.: The thermodynamics of computation—a review. Int. J. Theor. Phys. 21, 905–940 (1982)CrossRefGoogle Scholar
  47. 47.
    Fredkin, E., Toffoli, T.: Conservative logic. Int. J. Theor. Phys. 21, 219–253 (1982)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Deutsch, D.: Quantum theory, the Church Turing principle and the universal quantum computer. Proc. R. Soc. A 400, 97–117 (1985)ADSMathSciNetCrossRefGoogle Scholar
  49. 49.
    Aharonov, Y., Albert, D.Z., Vaidman, L.: How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100. Phys. Rev. Lett. 60, 1351 (1988)ADSCrossRefGoogle Scholar
  50. 50.
    Elitzur, A.C., Cohen, E., Okamoto, R., Takeuchi, S.: Nonlocal position changes of a photon revealed by quantum routers. Sci. Rep. 8, 7730 (2018)ADSCrossRefGoogle Scholar
  51. 51.
    Cohen, E., Pollak, E.: Determination of weak values of quantum operators using only strong measurements. Phys. Rev. A 98, 042112 (2018)ADSMathSciNetCrossRefGoogle Scholar
  52. 52.
    Bennett, C.H., Brassard, G.: Quantum cryptography: Public key distribution and coin tossing. In: Proceedings of IEEE international conference on computers, systems and signal processing, vol. 175, p. 8. New York (1984)Google Scholar
  53. 53.
    Ekert, A.K.: Quantum cryptography based on Bell’s theorem. Phys. Rev. Lett. 67, 661–663 (1991)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Elsag Bailey ICT Division and Quantum Information LaboratoryGenovaItaly
  2. 2.Faculty of Engineering and the Institute of Nanotechnology and Advanced MaterialsBar Ilan UniversityRamat GanIsrael
  3. 3.Mathematical InstituteUniversity of OxfordOxfordUK
  4. 4.Centre for Quantum TechnologiesNational University of SingaporeSingaporeSingapore
  5. 5.Institute for Quantum StudiesChapman UniversityOrangeUSA
  6. 6.Iyar, The Israeli Institute for Advanced ResearchZichron Ya’akovIsrael

Personalised recommendations