Interferometric Computation Beyond Quantum Theory

Part of the following topical collections:
  1. Foundational Aspects of Quantum Information


There are quantum solutions for computational problems that make use of interference at some stage in the algorithm. These stages can be mapped into the physical setting of a single particle travelling through a many-armed interferometer. There has been recent foundational interest in theories beyond quantum theory. Here, we present a generalized formulation of computation in the context of a many-armed interferometer, and explore how theories can differ from quantum theory and still perform distributed calculations in this set-up. We shall see that quaternionic quantum theory proves a suitable candidate, whereas box-world does not. We also find that a classical hidden variable model first presented by Spekkens (Phys Rev A 75(3): 32100, 2007) can also be used for this type of computation due to the epistemic restriction placed on the hidden variable.


Post-quantum computation Interference Deutsch–Jozsa algorithm Generalized probabilistic theories Mach–Zehnder interferometer Spekkens’ toy model 



The author is very grateful for illuminating discussions and correspondence with Felix Binder, Oscar Dahlsten, Daniela Frauchiger, Nana Liu, Markus Müller, Vlatko Vedral, and Benjamin Yadin. The author is grateful for financial support from the John Templeton Foundation and the Foundational Questions Institute. This research was undertaken whilst the author was funded by the Engineering and Physical Sciences Research Council (UK) at the University of Oxford. Some of concepts in this article have also appeared as part of the author’s DPhil thesis [31].


  1. 1.
    Deutsch, D.: Quantum theory, the church-turing principle and the universal quantum computer. Proc. R. Soc. A 400(1818), 97–117 (1985)ADSMathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Deutsch, D., Jozsa, R.: Rapid solution of problems by quantum computation. Proc. R. Soc. Lond. A. 439(1907), 553–558 (1992)ADSMathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Simon, D.R.: On the power of quantum computation. In: 35th Annual Symposium on Foundations of Computer Science, 1994 Proceedings, pp. 116–123 (1994)Google Scholar
  4. 4.
    Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, STOC’96, pp. 212–219, New York, NY. ACM (1996)Google Scholar
  5. 5.
    Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26(5), 1484–1509 (1997)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cleve, R., Ekert, A.K., Macchiavello, C., Mosca, M.: Quantum algorithms revisited. Proc. R. Soc. Lond. A. 454(1969), 339–354 (1998)ADSMathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Lloyd, S.: Quantum search without entanglement. Phys. Rev. A 61(1), 010301 (1999)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Zeilinger, A.: A foundational principle for quantum mechanics. Found. Phys. 29(4), 631–643 (1999)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hardy, L.: Quantum Theory from Five Reasonable Axioms (2001)Google Scholar
  10. 10.
    Fuchs, C.: Quantum Foundations in the Light of Quantum Information (2001). arXiv:quant-ph/0106166v1
  11. 11.
    Masanes, Ll, Müller, M.P.: A derivation of quantum theory from physical requirements. New J. Phys. 13(6), 63001 (2011)CrossRefGoogle Scholar
  12. 12.
    Chiribella, G., DAriano, G.M., Perinotti, P.: Quantum theory, namely the pure and reversible theory of information. Entropy 14(12), 1877–1893 (2012)ADSMathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Barnum, H., Müller, M.P., Ududec, C.: Higher-order interference and single-system postulates characterizing quantum theory. New J. Phys. 16(12), 123029 (2014)ADSCrossRefGoogle Scholar
  14. 14.
    Clauser, J., Horne, M., Shimony, A., Holt, R.: Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett. 23(15), 880–884 (1969)ADSCrossRefMATHGoogle Scholar
  15. 15.
    Cirel’son, B.S.: Quantum generalizations of Bell’s inequality. Lett. Math. Phys. 4(2), 93–100 (1980)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Tsirelson, B.S.: Some results and problems on quantum Bell-type inequalities. Hadron. J. Suppl. 8, 329–345 (1993)MathSciNetMATHGoogle Scholar
  17. 17.
    Popescu, S., Rohrlich, D.: Quantum nonlocality as an axiom. Found. Phys. 24(3), 379–385 (1994)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Popescu, S.: Nonlocality beyond quantum mechanics. Nat. Phys. 10(4), 264–270 (2014)CrossRefGoogle Scholar
  19. 19.
    van Dam, W.: Nonlocality & Communication Complexity. DPhil thesis, University of Oxford (2000)Google Scholar
  20. 20.
    Cleve, R., van Dam, W., Nielsen, M., Tapp, A.: Quantum entanglement and the communication complexity of the inner product function. Theor. Comput. Sci. 486, 11–19 (2013)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Fernandez, J.M., Schneeberger, W.A.: Quaternionic computing (2003). arXiv:quant-ph/0307017
  22. 22.
    Lee, C.M., Barrett, J.: Computation in generalised probabilistic theories. New J. Phys. 17, 083001 (2015)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Lee, C.M., Hoban, M.J.: Bounds on the power of proofs and advice in general physical theories. Proc. R. Soc. A 472(2190), 20160076 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Lee, C.M., Selby, J.H.: Generalised phase kick-back: the structure of computational algorithms from physical principles. New J. Phys. 18, 033023 (2016)ADSCrossRefGoogle Scholar
  25. 25.
    Lee, C.M., Selby, J.H.: Deriving Grover’s lower bound from simple physical principles. New J. Phys. 18, 093047 (2016)ADSCrossRefGoogle Scholar
  26. 26.
    Sorkin, R.: Quantum mechanics as quantum measure theory. Mod. Phys. Lett. A 9, 3119 (1994)ADSMathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Ududec, C., Barnum, H., Emerson, J.: Three slit experiments and the structure of quantum theory. Found. Phys. 41(3), 396–405 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Ududec, C.: Perspectives on the Formalism of Quantum Theory. PhD thesis, University of Waterloo (2012)Google Scholar
  29. 29.
    Garner, A.J.P., Dahlsten, O.C.O., Nakata, Y., Murao, M., Vedral, V.: A framework for phase and interference in generalized probabilistic theories. New J. Phys. 15(9), 093044 (2013)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Dahlsten, O.C.O., Garner, A.J.P., Vedral, V.: The uncertainty principle enables non-classical dynamics in an interferometer. Nat. Commun. 5, 4592 (2014)ADSCrossRefGoogle Scholar
  31. 31.
    Garner, A.J.P.: Phase and interference phenomena in generalized probabilistic theories. DPhil thesis, University of Oxford, Hilary Term (2015)Google Scholar
  32. 32.
    Spekkens, R.W.: Evidence for the epistemic view of quantum states: a toy theory. Phys. Rev. A 75(3), 32110 (2007)ADSCrossRefGoogle Scholar
  33. 33.
    Spekkens, R.W.: Quasi-quantization: classical statistical theories with an epistemic restriction. Quantum Theory, pp. 83–135 (2015)Google Scholar
  34. 34.
    Barrett, J.: Information processing in generalized probabilistic theories. Phys. Rev. A 75(3), 32304 (2007)ADSCrossRefGoogle Scholar
  35. 35.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  36. 36.
    Schrödinger, E.: Zum Heisenbergschen Unschärfeprinzip. S. B. Preuß. Akad. Wiss., Physikalisch-mathematische Klasse 14, 296–303 (1930)MATHGoogle Scholar
  37. 37.
    Robertson, H.P.: The uncertainty principle. Phys. Rev. 34(1), 163–164 (1929)ADSCrossRefGoogle Scholar
  38. 38.
    van Enk, S.J.: A toy model for quantum mechanics. Found. Phys. 37(10), 1447–1460 (2007)ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    Janotta, P., Lal, R.: Generalized probabilistic theories without the no-restriction hypothesis. Phys. Rev. A 87(5), 052131 (2013)ADSCrossRefGoogle Scholar
  40. 40.
    Pusey, Matthew F.: Stabilizer notation for Spekkens’ toy theory. Found. Phys. 42(5), 688–708 (2012)ADSMathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Aharonov, Y., Bohm, D.: Significance of electromagnetic potentials in the quantum theory. Phys. Rev. 115(3), 485–491 (1959)ADSMathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Gross, D., Müller, M., Colbeck, R., Dahlsten, O.C.O.: All reversible dynamics in maximally nonlocal theories are trivial. Phys. Rev. Lett. 104(8), 80402 (2010)ADSCrossRefGoogle Scholar
  43. 43.
    Dakic, B., Brukner, C.: The classical limit of a physical theory and the dimensionality of space (2013)Google Scholar
  44. 44.
    Müller, M.P., Masanes, Ll: Three-dimensionality of space and the quantum bit: an information-theoretic approach. New J. Phys. 15(5), 053040 (2013)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Garner, A.J.P., Müller, M.P., Dahlsten, O.C.O.: The complex and quaternionic quantum bit from relativity of simultaneity on an interferometer. Proc. R. Soc. A 473(2208), 20170596 (2017).
  46. 46.
    Alfsen, E.M., Shultz, F.W.: Geometry of State Spaces of Operator Algebras. Birkhäuser, Boston (2003)CrossRefMATHGoogle Scholar
  47. 47.
    Dickson, L.E.: On quaternions and their generalization and the history of the eight square theorem. Ann. Math. Second Series 20(3), 155–171 (1919)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Graydon, M.A.: Quaternions and quantum theory. Master’s thesis, University of Waterloo (2011)Google Scholar
  49. 49.
    Baez, J.C.: Symplectic, Quaternionic, Fermionic (2014).
  50. 50.
    Bohm, D.: A suggested interpretation of the quantum theory in terms of “hidden” variables. I. Phys. Rev. 85(2), 166–179 (1952a)ADSMathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Bohm, D.: A suggested interpretation of the quantum theory in terms of “hidden” variables. II. Phys. Rev. 85(2), 180–193 (1952b)ADSMathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Norsen, Travis: The pilot-wave perspective on spin. Am. J. Phys. 82(4), 337–348 (2014)ADSCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Centre for Quantum TechnologiesNational University of SingaporeSingaporeSingapore

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