Advertisement

Implementation of quantum secret sharing and quantum binary voting protocol in the IBM quantum computer

  • Dintomon JoyEmail author
  • M. Sabir
  • Bikash K. Behera
  • Prasanta K. Panigrahi
Article
First Online:
  • 62 Downloads

Abstract

Quantum secret sharing is a way to share secret messages among the clients in a group with complete security. For the first time, Hillery et al. (Phys Rev A 59:1829, 1999) proposed the quantum version of classical secret sharing protocol using GHZ states. Here, we implement the above quantum secret sharing protocol in ‘IBM Q 5 Tenerife’ quantum processor and compare the experimentally obtained results with the theoretically predicted ones. Further, a new quantum binary voting protocol is proposed and implemented in the 14-qubit ‘IBM Q 14 Melbourne’ quantum processor. The results are analyzed through the technique of quantum state tomography, and the fidelity of states is calculated for different number of executions made in the device.

Keywords

Quantum secret sharing Quantum binary voting Quantum state tomography IBM Q 5 Tenerife IBM Q 14 Melbourne 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

D.J. thanks U.G.C., New Delhi, for providing the financial support through BSR fellowship and Athul R.T. for the technical support. B.K.B. acknowledges the financial support of IISER Kolkata. The authors are extremely grateful to IBM team and IBM QE project. The discussions and opinions developed in this paper are only those of the authors and do not reflect the opinions of IBM or IBM Quantum Experience team.

References

  1. 1.
    Shamir, A.: How to share a secret. Commun. ACM 22, 612 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Blakley, G.R.: Safeguarding cryptographic keys. In: International Workshop on Managing Requirements Knowledge, pp. 313–317. IEEE Computer Society, New York (1979)Google Scholar
  3. 3.
    Gottesman, D.: Theory of quantum secret sharing. Phys. Rev. A 61, 042311 (2000)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Singh, S.K., Srikanth, R.: Generalized quantum secret sharing. Phys. Rev. A 71, 012328 (2005)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Schneier, B.: Applied Cryptography, p. 70. Wiley, New York (1996)zbMATHGoogle Scholar
  6. 6.
    Gruska, J.: Foundations of Computing, p. 504. Thomson Computer Press, London (1997)Google Scholar
  7. 7.
    Shor, P.W.: Algorithms for quantum computation: Discrete logarithms and factoring. In: Proceedings 35th Annual Symposium on Foundations of Computer Science, pp. 124–134. Santa Fe, New Mexico (1994)Google Scholar
  8. 8.
    Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proceedings of the 28th Annual ACM Symposium Theory of Computing, pp. 212–219. ACM Press, New York (1996)Google Scholar
  9. 9.
    Gangopadhyay, S., Behera, B.K., Panigrahi, P.K.: Generalization and demonstration of an entanglement-based Deutsch–Jozsa-like algorithm using a 5-qubit quantum computer. Quantum Inf. Process. 17, 160 (2018)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Wang, J., Li, L., Peng, H., Yang, Y.: Quantum-secret-sharing scheme based on local distinguishability of orthogonal multiqudit entangled states. Phys. Rev. A 95, 022320 (2017)ADSCrossRefGoogle Scholar
  11. 11.
    Hillery, M., Bužek, V., Berthiaume, A.: Quantum secret sharing. Phys. Rev. A 59, 1829 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Wiesner, S.: Conjugate coding. ACM SIGACT News 15, 78 (1983) CrossRefzbMATHGoogle Scholar
  13. 13.
    Wang, Z.Y., Yuan, H., Shi, S.H., Zhang, Z.J.: Three-party qutrit-state sharing. Eur. Phys. J. D 41, 371 (2007)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Cleve, R., Gottesman, D., Lo, H.K.: How to share a quantum secret. Phys. Rev. Lett. 83, 648 (1999)ADSCrossRefGoogle Scholar
  15. 15.
    Matsumoto, R.: Unitary reconstruction of secret for stabilizer-based quantum secret sharing. Quantum Inf. Process. 16, 202 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lu, H., et al.: Secret sharing of a quantum state. Phys. Rev. Lett. 117, 030501 (2016)ADSCrossRefGoogle Scholar
  17. 17.
    Sarvepalli, P.: Nonthreshold quantum secret-sharing schemes in the graph-state formalism. Phys. Rev. A 86, 042303 (2012)ADSCrossRefGoogle Scholar
  18. 18.
    Gravier, S., Javelle, J., Mhalla, M., Perdrix, S.: On weak odd domination and graph-based quantum secret sharing. Theor. Comput. Sci. 598, 129 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Diep, D.N., Giang, D.H., Phu, P.H.: Application of quantum Gauss–Jordan elimination code to quantum secret sharing code. Int. J. Theor. Phys. 57, 841 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Gao, G., Wang, Y., Wang, D., Ye, L.: Comment on ’Authenticated quantum secret sharing with quantum dialogue based on Bell states’. Phys. Scr. 93, 027002 (2018)ADSCrossRefGoogle Scholar
  21. 21.
    Abulkasim, H., Hamad, S., Elhadad, A.: Reply to Comment on ’Authenticated quantum secret sharing with quantum dialogue based on Bell states’. Phys. Scr. 93, 027001 (2018)ADSCrossRefGoogle Scholar
  22. 22.
    Liu, F., Qin, S.J., Wen, Q.Y.: A quantum secret-sharing protocol with fairness. Phys. Scr. 89, 075104 (2014)ADSCrossRefGoogle Scholar
  23. 23.
    Lai, H., Xiao, J., Orgun, M.A., Xue, L., Pieprzyk, J.: Quantum direct secret sharing with efficient eavesdropping-check and authentication based on distributed fountain codes. Quantum Inf. Process. 13, 895 (2014)ADSCrossRefzbMATHGoogle Scholar
  24. 24.
    Xie, C., Li, L., Qiu, D.: A novel semi-quantum secret sharing scheme of specific bits. Int. J. Theor. Phys. 54, 3819 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Li, L., Qiu, D., Mateus, P.: Quantum secret sharing with classical Bobs. J. Phys. A Math. Theor. 46, 045304 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Tsai, C., Hwang, T.: Multi-party quantum secret sharing based on two special entangled states. Sci. China Phys. Mech. Astron. 55, 460 (2012)ADSCrossRefGoogle Scholar
  27. 27.
    Helwig, W., Cui, W., Latorre, J.I., Riera, A., Lo, H.K.: Absolute maximal entanglement and quantum secret sharing. Phys. Rev. A 86, 052335 (2012)ADSCrossRefGoogle Scholar
  28. 28.
    Liao, C.H., Yang, C.W., Hwang, T.: Dynamic quantum secret sharing protocol based on GHZ state. Quantum Inf. Process. 13, 1907 (2014)ADSCrossRefzbMATHGoogle Scholar
  29. 29.
    Gao, G.: Improvement of efficient multiparty quantum secret sharing based on bell states and continuous variable operations. Int. J. Theor. Phys. 53, 2231 (2014)CrossRefGoogle Scholar
  30. 30.
    Zhang, L., Guo, Y., Huang, D.: High-efficient quantum secret sharing with arrangements of lines on two-dimensional planes. Int. J. Internet Protoc. Technol. 8, 116 (2014)CrossRefGoogle Scholar
  31. 31.
    Zhang, Z.J., Man, Z.X.: Multiparty quantum secret sharing of classical messages based on entanglement swapping. Phys. Rev. A 72, 022303 (2005)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    Massoud, H.D., Elham, F.: A novel and efficient multiparty quantum secret sharing scheme using entangled states. Sci. China Phys. Mech. Astron. 55, 1828 (2012)ADSCrossRefGoogle Scholar
  33. 33.
    Guo, G.P., Guo, G.C.: Quantum secret sharing without entanglement. Phys. Lett. A 310, 247 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Hsu, L.Y., Li, C.M.: Quantum secret sharing using product states. Phys. Rev. A 71, 022321 (2005)ADSCrossRefGoogle Scholar
  35. 35.
    Yan, F., Gao, T.: Quantum secret sharing between multiparty and multiparty without entanglement. Phys. Rev. A 72, 012304 (2005)ADSCrossRefGoogle Scholar
  36. 36.
    Han, L.F., Liu, Y.M., Liu, J., Zhang, Z.J.: Multiparty quantum secret sharing of secure direct communication using single photons. Opt. Commun. 281, 2690 (2008)ADSCrossRefGoogle Scholar
  37. 37.
    Zhang, Z.J., Li, Y., Man, Z.X.: Multiparty quantum secret sharing. Phys. Rev. A 71, 044301 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Chen, X.B., Niu, X.X., Zhou, X.J., Yang, Y.X.: Multi-party quantum secret sharing with the single-particle quantum state to encode the information. Quantum Inf. Process. 12, 365 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Wang, H., Huang, Y., Fang, X., Gu, B., Fu, D.: High-capacity three-party quantum secret sharing with single photons in both the polarization and the spatial-mode degrees of freedom. Int. J. Theor. Phys. 52, 1043 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Wang, T.Y., Wen, Q.Y., Chen, X.B., Guo, F.Z., Zhu, F.C.: An efficient and secure multiparty quantum secret sharing scheme based on single photons. Opt. Commun. 281, 6130 (2008)ADSCrossRefGoogle Scholar
  41. 41.
    Hao, S.B., Yu, B.: Multiparty quantum secret information sharing in enterprise management based on single qubit with random rotation angle. Int. J. Theor. Phys. 51, 1674 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Muralidharan, S., Panigrahi, P.K.: Perfect teleportation, quantum-state sharing, and superdense coding through a genuinely entangled five-qubit state. Phys. Rev. A 77, 032321 (2008)ADSCrossRefGoogle Scholar
  43. 43.
    Muralidharan, S., Panigrahi, P.K.: Quantum-information splitting using multipartite cluster states. Phys. Rev. A 78, 062333 (2008)ADSCrossRefGoogle Scholar
  44. 44.
    Choudhury, S., Muralidharan, S., Panigrahi, P.K.: Quantum teleportation and state sharing using a genuinely entangled six-qubit state. J. Phys. A: Math. Theor. 42, 115303 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Hou, K., Li, Y.B., Shi, S.H.: Quantum state sharing with a genuinely entangled five-qubit state and Bell-state measurements. Opt. Commun. 283, 1961 (2010)ADSCrossRefGoogle Scholar
  46. 46.
    Li, Y.H., Liu, J.C., Nie, Y.Y.: Quantum teleportation and quantum information splitting by using a genuinely entangled six-qubit state. Int. J. Theor. Phys. 49, 2592 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Nie, Y.Y., Li, Y.H., Liu, J.C., Sang, M.H.: Quantum information splitting of an arbitrary three-qubit state by using two four-qubit cluster states. Quantum Inf. Process. 10, 297 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Nie, Y.Y., Li, Y.H., Liu, J.C., Sang, M.H.: Quantum state sharing of an arbitrary three-qubit state by using four sets of W-class states. Opt. Commun. 284, 1457 (2011)ADSCrossRefGoogle Scholar
  49. 49.
    Muralidharan, S., Karumanchi, S., Narayanaswamy, S., Srikanth, R., Panigrahi, P.K.: In How Many Ways Can Quantum Information Be Split? arXiv:0907.3532
  50. 50.
    Tittel, W., Zbinden, H., Gisin, N.: Experimental demonstration of quantum secret sharing. Phys. Rev. A 63, 042301 (2011)ADSCrossRefGoogle Scholar
  51. 51.
    Schmid, C., Trojek, P., Bourennane, M., Kurtsiefer, C., Zukowski, M., Weinfurter, H.: Experimental single qubit quantum secret sharing. Phys. Rev. Lett. 95, 230505 (2005)ADSCrossRefGoogle Scholar
  52. 52.
    Gaertner, S., Kurtsiefer, C., Bourennane, M., Weinfurter, H.: Experimental demonstration of four-party quantum secret sharing. Phys. Rev. Lett. 98, 020503 (2007)ADSCrossRefGoogle Scholar
  53. 53.
    Wei, K.J., Ma, H.Q., Yang, J.H.: Experimental circular quantum secret sharing over telecom fiber network. Opt. Express 21, 16663 (2013)ADSCrossRefGoogle Scholar
  54. 54.
    IBM—Quantum Network. https://www.research.ibm.com/ibm-q/network/overview/. Accessed 18 July 2019
  55. 55.
    Boixo, S., et al.: Characterizing quantum supremacy in near-term devices. Nat. Phys. 14, 595–600 (2018)CrossRefGoogle Scholar
  56. 56.
    Connover, E.: Google moves toward quantum supremacy with 72-qubit computer. Sci. News 193, 6 (2018)Google Scholar
  57. 57.
    Gibney, E.: Inside Microsoft’s quest for a topological quantum computer. Nat. News (2016).  https://doi.org/10.1038/nature.2016.20774 CrossRefGoogle Scholar
  58. 58.
    Microsoft Quantum Development Kit. https://microsoft.com/quantum (2017). Accessed 18 July 2019
  59. 59.
    The Future of Quantum Computing is Counted in Qubits. https://newsroom.intel.com/news/future-quantum-computing-counted-qubits/ (2018). Accessed 18 July 2019
  60. 60.
    Rigetti, C.: The Rigetti 128-qubit chip and what it means for quantum, Medium https://medium.com/rigetti/the-rigetti-128-qubit-chip-and-what-it-means-for-quantum-df757d1b71ea (2018). Accessed 18 July 2019
  61. 61.
    D-Wave Announces D-Wave 2000Q Quantum Computer and First System Order, press release. https://www.dwavesys.com/press-releases/d-wave%C2%A0announces%C2%A0d-wave-2000q-quantum-computer-and-first-system-order (2017). Accessed 18 July 2019
  62. 62.
    Neukart, F., Compostella, G., Seidel, C., Von Dollen, D., Yarkoni, S., Parney, B.: Traffic flow optimization using a quantum annealer. Front. ICT 4, 29 (2017).  https://doi.org/10.3389/fict.2017.00029 CrossRefGoogle Scholar
  63. 63.
    Boyda, E., Basu, S., Ganguly, S., Michaelis, A., Mukhopadhyay, S., Nemani, R.R.: Deploying a quantum annealing processor to detect tree cover in aerial imagery of California. PLoS ONE 12(2), e0172505 (2017)CrossRefGoogle Scholar
  64. 64.
    O’Malley, D., Vesselinov, V.V., Alexandrov, B.S., Alexandrov, L.B.: Nonnegative/binary matrix factorization with a D-Wave quantum annealer. PLoS ONE 13(12), e0206653 (2018)CrossRefGoogle Scholar
  65. 65.
    Srivastava, R., Choi, I., Cook, T.: The Commercial Prospects of Quantum Computing. Technical report, Networked Quantum Information Technologies (2016)Google Scholar
  66. 66.
    Gabriel, P.: Quest for qubits. Science 354(6316), 1090–1093 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  67. 67.
    Cusumano, M.A.: The business of quantum computing. Commun. ACM 61(10), 20–22 (2018)CrossRefGoogle Scholar
  68. 68.
    Fedortchenko, S.: A Quantum Teleportation Experiment for Undergraduate Students (2016). arXiv:1607.02398
  69. 69.
    Sisodia, M., Shukla, A., Thapliyal, K., Pathak, A.: Design and experimental realization of an optimal scheme for teleportation of an n-qubit quantum state. Quantum Inf. Process. 16, 292 (2017)ADSMathSciNetCrossRefGoogle Scholar
  70. 70.
    Alsina, D., Latorre, J.I.: Experimental test of Mermin inequalities on a five-qubit quantum computer. Phys. Rev. A 94, 012314 (2016)ADSCrossRefGoogle Scholar
  71. 71.
    Berta, M., Wehner, S., Wilde, M.M.: Entropic uncertainty and measurement reversibility. New J. Phys. 18, 073004 (2016)ADSCrossRefGoogle Scholar
  72. 72.
    Devitt, S.J.: Performing quantum computing experiments in the cloud. Phys. Rev. A 94, 032329 (2016)ADSCrossRefGoogle Scholar
  73. 73.
    Takita, M., Corcoles, A.D., Magesan, E., Abdo, B., Brink, M., Cross, A., Chow, J.M., Gambetta, J.M.: Demonstration of weight-four parity measurements in the surface code architecture. Phys. Rev. Lett. 117, 210505 (2016)ADSCrossRefGoogle Scholar
  74. 74.
    Wootton, J.R., Loss, D.: Repetition code of 15 qubits. Phys. Rev. A 97, 052313 (2018)ADSCrossRefGoogle Scholar
  75. 75.
    Ghosh, D., Agarwal, P., Pandey, P., Behera, B.K., Panigrahi, P.K.: Automated error correction in IBM quantum computer and explicit generalization. Quantum Inf. Process. 17, 153 (2018)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  76. 76.
    Behera, B.K., Banerjee, A., Panigrahi, P.K.: Experimental realization of quantum cheque using a five-qubit quantum computer. Quantum Inf. Process. 16, 312 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  77. 77.
    Sisodia, M., Shukla, A., Pathak, A.: Experimental realization of nondestructive discrimination of Bell states using a five-qubit quantum computer. Phys. Lett. A 381, 3860 (2017)ADSCrossRefGoogle Scholar
  78. 78.
    Vuillot, C.: Is Error Detection Helpful on IBM 5Q Chips? (2018). arXiv:1705.08957v1
  79. 79.
    Huang, H.L., Zhao, Y.W., Li, T., Li, F.G., Du, Y.T., Fu, X.Q., Zhang, S., Wang, X., Bao, W.S.: Homomorphic encryption experiments on IBM’s cloud quantum computing platform. Front. Phys. 12, 120305 (2017)CrossRefGoogle Scholar
  80. 80.
    Wooton, J.R.: Demonstrating non-Abelian braiding of surface code defects in a five qubit experiment. Quantum Sci. Technol. 2, 015006 (2017)ADSCrossRefGoogle Scholar
  81. 81.
    Li, R., Alvarez-Rodriguez, U., Lamata, L., Solano, E.: Approximate quantum adders with genetic algorithms: an IBM quantum experience. Quantum Meas. Quantum Metrol. 4, 1 (2017)ADSCrossRefGoogle Scholar
  82. 82.
    Deffner, S.: Demonstration of entanglement assisted invariance on IBM’s quantum experience. Heliyon 3, e00444 (2017)CrossRefGoogle Scholar
  83. 83.
    Hebenstreit, M., Alsina, D., Latorre, J.I., Kraus, B.: Compressed quantum computation using a remote five-qubit quantum computer. Phys. Rev. A 95, 052339 (2017)ADSCrossRefGoogle Scholar
  84. 84.
    Linke, N.M., Maslov, D., Roetteler, M., Debnath, S., Figgatt, C., Landsman, K.A., Wright, K., Monroe, C.: Experimental comparison of two quantum computing architectures. Proc. Natl. Acad. Sci. 114, 3305 (2017)CrossRefGoogle Scholar
  85. 85.
    Hillery, M., Ziman, M., Bužek, V., Bieliková, M.: Towards quantum-based privacy and voting. Phys. Lett. A 349, 75–81 (2006)ADSCrossRefzbMATHGoogle Scholar
  86. 86.
    Sharma, R.D., De, A.: Quantum voting using single qubits. Indian J. Sci. Technol. 9(42), 032329 (2016)CrossRefGoogle Scholar
  87. 87.
    Ghose, S., Kumar, A., Madhok, V., Hamel, A.M.: Multiparty quantum communication using multiqubit entanglement and teleportation. Phys. Res. Int. 2014, 948750 (2014)CrossRefGoogle Scholar
  88. 88.
    Tian, J.H., Zhang, J.Z., Li, Y.P.: A voting protocol based on the controlled quantum operation teleportation. Int. J. Theor. Phys. 55, 2303 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  89. 89.
    Thapliyal, K., Sharma, R.D., Pathak, A.: Protocols for quantum binary voting. Int. J. Quantum Inf. 15(1), 1750007 (2017)CrossRefzbMATHGoogle Scholar
  90. 90.
    Lou, X.: Quantum distributed ballot scheme based on entanglement swapping. In: 10th International Conference on Trust. Security and Privacy in Computing and Communications, IEEE (2011)Google Scholar
  91. 91.
    Cao, H.J., Ding, L.Y., Jiang, X.L., Li, P.F.: A new proxy electronic voting scheme achieved by six-particle entangled states. Int. J. Theor. Phys. 57, 674 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  92. 92.
    Borras, A., Plastino, A.R., Batle, J., Zander, C., Casas, M., Plastino, A.: Multiqubit systems: highly entangled states and entanglement distribution. J. Phys. A Math. Theor. 40, 13407 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  93. 93.
    Zhang, J.L., Xie, S.C., Zhang, J.Z.: An elaborate secure quantum voting scheme. Int. J. Theor. Phys. 56, 3019 (2018)CrossRefzbMATHGoogle Scholar
  94. 94.
    Xue, P., Zhang, X.: A simple quantum voting scheme with multi-qubit entanglement. Sci. Rep. 7, 7586 (2017)ADSCrossRefGoogle Scholar
  95. 95.
    Karlsson, A., Bourennane, M.: Quantum teleportation using three-particle entanglement. Phys. Rev. A 58, 4394 (1998)ADSMathSciNetCrossRefGoogle Scholar
  96. 96.
    Bennett, C.H., Brassard, G., Crépeau, C., Jozsa, R., Peres, A., Wootters, W.K.: Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels. Phys. Rev. Lett. 70, 1895 (1993)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  97. 97.
    Wootters, W.K., Zurek, W.H.: A single quantum cannot be cloned. Nature 299, 802–803 (1982)ADSCrossRefzbMATHGoogle Scholar
  98. 98.
    Benenti, G., Casati, G., Strini, G.: Principles of Quantum Computation and Information-Volume I: Basic Concepts, p. 118. World scientific, Singapore (2004)CrossRefzbMATHGoogle Scholar
  99. 99.
    Pathak, A.: Experimental Quantum Mechanics in the Class Room: Testing Basic Ideas of Quantum Mechanics and Quantum Computing Using IBM Quantum Computer (2018). arXiv:1805.06275v1
  100. 100.
    Altepeter, J.B., James, D.F.V., Kwiat, P.G.: Quantum state tomography. In: Paris, M., Rehácek, J. (eds.) Quantum State Estimation. Lecture Notes in Physics, vol. 649. Springer, Berlin (2004) Google Scholar
  101. 101.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information, p. 409. Cambridge University Press, Cambridge (2010)CrossRefGoogle Scholar
  102. 102.
    Zhang, J.L., Zhan, J.Z., Xie, S.C.: A choreographed distributed electronic voting scheme. Int. J. Theor. Phys. 57, 2676 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  103. 103.
    Vaccaro, J.A., Spring, J., Chefles, A.: Quantum protocols for anonymous voting and surveying. Phys. Rev. A 75, 012333 (2007)ADSCrossRefGoogle Scholar
  104. 104.
    Sun, X., Wang, Q., Kulicki, P.: A simple voting protocol on quantum blockchain. Int. J. Theor. Phys. 58, 275 (2018)CrossRefzbMATHGoogle Scholar
  105. 105.
    Zhou, N., Zeng, G., Zeng, W., Zhu, F.: Cross-center quantum identification scheme based on teleportation and entanglement swapping. Opt. Commun. 254, 380 (2005)ADSCrossRefGoogle Scholar
  106. 106.
    Cabello, A.: Quantum key distribution in the Holevo limit. Phys. Rev. Lett. 85, 5635 (2000)ADSCrossRefGoogle Scholar
  107. 107.
    Banerjee, A., Pathak, A.: Maximally efficient protocols for direct secure quantum communication. Phys. Lett. A 376, 2944–2950 (2012)ADSCrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of PhysicsCochin University of Science and TechnologyKochiIndia
  2. 2.Department of Physical SciencesIndian Institute of Science Education and Research KolkataMohanpurIndia

Personalised recommendations

Cite article

Buy options