Topological quantum structures from association schemes

Abstract

Starting from an association scheme induced by a finite group and the corresponding Bose–Mesner algebra, we construct quantum Markov chains, their entangled versions, using the quantum probabilistic approach. Our constructions are based on the intersection numbers and their duals Krein parameters of the schemes. We make the connection for the first time between the fusion rules of anyonic particles evolving on a 2D surface to the Krein parameters of an association scheme. We consider braid group \(B_3\) that describes the unitary dynamics of the anyons as the automorphism subgroup of the graphs. The dynamics induced by the fusions (and the adjoint splitting operations) may be viewed as the chain evolving on a growing graph and the braiding as automorphisms on a fixed graph. In our quantum probability framework, infinite iterations of the unitaries, which can encode algorithmic content for quantum simulations, can describe asymptotics elegantly if the particles are allowed to evolve coherently for a longer period. We define quantum states on the Bose–Mesner algebra which is also a von Neumann algebra as well as a Frobenius algebra to build the quantum Markov chains providing yet another perspective to topological computation, whereas frameworks such as Unitary Modular Categories can identify and characterize new anyonic systems our framework can build upon them within quantum probabilistic framework that are suitable for asymptotic analysis.

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References

  1. 1.

    Bratteli, O., Jorgensen, P.E.T., Kishimoto, A., Werner, R.F.: Pure states on \({\cal{O} }_d\). J. Oper. Theory 43, 97–143 (2000)

    Google Scholar 

  2. 2.

    Ph. Biane.: Marches de Bernoulli quantiques, Universit  de Paris VII, preprint (1989)

  3. 3.

    Parthasarathy, K.R.: A generalized Biane Process. Lect. Not. Math. 1426, 345 (1990)

    Article  Google Scholar 

  4. 4.

    Olmschenk, S., Matsukevich, D.N., Maunz, P., Hayes, D., Duan, L.-M., Monroe, C.: Quantum teleportation between distant matter qubits. Science 323, 486 (2009)

    ADS  Article  Google Scholar 

  5. 5.

    Zieschang, P.-H.: Theory of Association Schemes. Springer, Berlin (2005b)

    Google Scholar 

  6. 6.

    Bailey, R.A.: Schemes, Association: Designed Experiments. Algebra and Combinatorics. Cambridge University Press, Cambridge (2004)

    Google Scholar 

  7. 7.

    Balu, R., Castillo, D., Siopsis, G.: Physical realization of topological quantum walks on IBM-Q and beyond. Quant. Sci. Tech. 3(3), 1 (2018)

    Google Scholar 

  8. 8.

    Balu, R.: Quantum Structures from Association Schemes. arXiv:1902.08664

  9. 9.

    Turaev, V.G.: Quantum Invariants of Knots and 3-Manifolds, De Gruyter Studies in Mathematics, vol. 18. Walter de Gruyter and Co., Berlin (1994)

    Google Scholar 

  10. 10.

    Freed, D.S.: The cobordism hypothesis. Bull. Am. Math. Soc. (N.S.) 50(1), 57–92 (2013). MR2994995

    MathSciNet  Article  Google Scholar 

  11. 11.

    Wang, Z.: Quantum Computing: A Quantum Group Approach, Symmetries and Groups in Contemporary Physics. In: Nankai Ser. Pure Appl. Math. Theoret. Phys., vol. 11, pp. 41–50. World Sci. Publ., Hackensack, NJ (2013)

  12. 12.

    Connes, A., Consani, C.: The hyperring of adele classes. J. Number Theory 131(2), 159–194 (2011)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Completely Bounded Maps and Operator Algebras. Cambridge Studies in Advanced Mathematics, vol. 78. Press Syndicate of the University of Cambridge, Cambridge (2002)

  14. 14.

    L’evy, T.: Topological quantum field theories and Markovian random fields. Bull. Sci. Math. 135(6–7), 629–649 (2011)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Accardi, L., Lu, Y.G., Volovich, I.: Quantum Theory and its Stochastic Limit. Springer, Berlin (2002)

    Google Scholar 

  16. 16.

    Santra, S., Balu, R.: Propagation of correlations in local random circuits. Quant. Info. Proc. 15, 4613 (2016)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, Cambridge (1995)

    Google Scholar 

  18. 18.

    Szegedy, M.: Quantum Speed-Up of Markov Chain Based Algorithms. In: Proceedings of 45th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 32–41. IEEE (2004)

  19. 19.

    Balu, R., Liu, C., Venegas-Andraca, S.: Probability distributions for Markov chains based quantum walks. J. Phys. A: Math. Theor. (2017)

  20. 20.

    Accardi, L., Fidaleo, F.: Entangled Markov chains. Ann. Mat. Pura Appl (2004)

  21. 21.

    Fannes, M., Nahtergaele, B., Werner, R.F.: Finitely correlated pure states. J. Funct. Anal. 120, 511 (1992)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Accardi, L., Matsuoka, T., Ohya, M.: Entanglcd Markov chains are indeed entangled. Infinit. Dimens. Anal. Quant. Probab. Rel. Top. 9, 379–390 (2006)

    Article  Google Scholar 

  23. 23.

    Tezak, N., Niederberger, A., Pavlichin, D.S., Sarma, G., Mabuchi, H.: Specification of photonic circuits using quantum hardware description language. Philos. Trans. A 370, 5270 (2012)

    ADS  MathSciNet  Article  Google Scholar 

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Acknowledgements

The author is grateful to Joseph W.Iverson (jiverson@math.umd.edu) for introducing the fascinating topic of association schemes and acknowledges his contribution to the first result.

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Correspondence to Radhakrishnan Balu.

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Balu, R. Topological quantum structures from association schemes. Quantum Inf Process 20, 42 (2021). https://doi.org/10.1007/s11128-020-02931-y

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Keywords

  • Association schemes
  • Bose–Mesner algebras
  • Anyonic computation
  • Enatnagled quantum Markov chains