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Quantum Walk Search on a Two-dimensional Grid with Extra Edges

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Abstract

Quantum walk has been successfully used to search for targets on graphs with vertices identified as the elements of a database. This spacial search on a two-dimensional periodic grid takes \(\mathcal {O}\left( \sqrt{N\log N}\right) \) oracle consultations to find a target vertex from N number of vertices with \(\mathcal {O}(1)\) success probability, while reaching optimal speed of \(\mathcal {O}(\sqrt{N})\) on \(d \ge 3\) dimensional square lattice. Our numerical analysis based on lackadaisical quantum walks searches M vertices on a 2-dimensional grid with optimal speed of \(\mathcal {O}(\sqrt{N/M})\), provided the grid is attached with additional long range edges. Based on the numerical analysis performed with multiple sets of randomly generated targets for a wide range of N and M we suggest that the optimal time complexity of \(\mathcal {O}(\sqrt{N/M})\) with constant success probability can be achieved for quantum search on a two-dimensional periodic grid with long-range edges.

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Data Availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

References

  1. Preskill, J.: Quantum 2, 79 (2018)

    Article  Google Scholar 

  2. Nielsen, M.A., Chuang, I.L.: Quantum computation and quantum information. Cambridge University Press, Cambridge (2000)

  3. Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. Proc: 35th Annual Symposium on Foundations of Computer Science, Santa Fe, NM, 20-22, Nov. (1994)

  4. Shor, P.W.: SIAM: J. Sci. Stat. Comput. 26, 1484 (1997)

  5. Grover, L.K.: A fast quantum mechanical algorithm for database search, vol. 212. In: Proceeding of 28th Annual ACM Symposium on Theory of Computing (STOC). (1996)

  6. Grover, L.K.: Phys. Rev. Lett. 79, 325 (1997)

    Article  ADS  Google Scholar 

  7. Grover, L.K., Radhakrishnan, J., Symp, A.C.M.: Parallel Algorithms and Architectures, Las Vegas, vol. 186. Nevada, USA (2005)

    Google Scholar 

  8. Choi, B.S., Korepin, V.E.: Quantum Inf. Process. 6, 37 (2007)

    Article  MathSciNet  Google Scholar 

  9. Zhang, K., Korepin, V.E.: Quantum Inf. Process. 17, 143 (2018)

    Article  ADS  Google Scholar 

  10. Giri, P.R., Korepin, V.E.: Quant. Inf. Process. 16, 1–36 (2017)

    Article  Google Scholar 

  11. Childs, A.M., Goldstone, J.: Phys. Rev. A 70, 022314 (2004)

    Article  ADS  Google Scholar 

  12. Aaronson, S., Ambainis, A.: Theor. Comput. 1(4), 47–79 (2005)

    Article  Google Scholar 

  13. Ambainis, A., Kempe, J., Rivosh, A.: Proceedings 16th Annual ACM-SIAM Symposium Discrete Algorithms, SODA 05. SIAM, Philadelphia, PA, pp. 1099–1108 (2005)

  14. Meyer, D.A., Wong, T.G.: Phys. Rev. Lett. 114, 110503 (2015)

    Article  ADS  Google Scholar 

  15. Chakraborty, S., Novo, L., Ambainis, A., Omar, Y.: Phys. Rev. Lett. 116, 100501 (2016)

  16. Benioff, P.: Contemporary Mathematics 305 112. American Mathematical Society, Providence, RI (2002)

  17. Brassard, G., Høyer, P., Mosca, M., Tapp, A.: Quantum amplitude amplification and estimation. Quantum Computation and Quantum Information 305, 53–74 (2000)

  18. Portugal, R.: Quantum walks and search algorithms. Springer, New York (2013)

    Book  MATH  Google Scholar 

  19. Childs, A.M., Goldstone, J.: Phys. Rev. A 70, 042312 (2004)

    Article  MathSciNet  ADS  Google Scholar 

  20. Tulsi, A.: Phys. Rev. A 78, 012310 (2008)

    Article  ADS  Google Scholar 

  21. Ambainis, A., Bačkurs, A., Nahimovs, N., Ozols, R., Rivosh, A.: Proceedings 7th Annual Conference Theory of Quantum Computation, Communication, and Cryptography, TQC 2012, 87-97. Springer, Tokyo (2013)

  22. Wong, T.G.: J. Phys. A Math. Theor. 48(43), 435304 (2015)

    Article  Google Scholar 

  23. Wong, T.G.: J. Phys. A Math. Theor. 50(47), 475301 (2017)

  24. Wong, T.G.: Quant. Inf. Process. 17, 68 (2018)

    Article  ADS  Google Scholar 

  25. Nahimovs, N., Rivosh, A.: Proceedings of SOFSEM 9587, 381–391 (2016)

    Google Scholar 

  26. Saha, A., Majumdar, R., Saha, D., Chakrabarti, A., Sur-Kolay, S.: Quant. Inf. Process 21, 275 (2022). Preprint at http://arxiv.org/abs/1804.01446 [quant-ph]

  27. Nahimovs, N.: SOFSEM 2019: Theory and practice of computer science. SOFSEM 2019. Lecture Notes in Computer Science, vol. 11376. Springer, Cham (2019)

  28. Giri, P.R., Korepin, V.E.: Mod. Phys. Lett. A 33(1), 2050043 (2020)

  29. Boettcher, S., Goncalves, B.: Europhys. Lett. 84, 30002 (2008)

    Article  ADS  Google Scholar 

  30. Marquezino, F.L., Portugal, R., Boettcher, S.: Phys. Rev. A 87, 012329 (2013)

    Article  ADS  Google Scholar 

  31. Giri, P.R., Korepin, V.E.: Int. J. Quant. Inf. 17(07), 1950060 (2019)

  32. Zalka, C.: Phys. Rev. A 60, 2746 (1999)

    Article  ADS  Google Scholar 

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  1. KDDI Research, Inc., Fujimino-shi, Saitama, Japan

    Pulak Ranjan Giri

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Correspondence to Pulak Ranjan Giri.

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Giri, P.R. Quantum Walk Search on a Two-dimensional Grid with Extra Edges. Int J Theor Phys 62, 121 (2023). https://doi.org/10.1007/s10773-023-05369-x

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