Abstract
The lackadaisical quantum walk is a graph search algorithm for 2D grids whose vertices have a self-loop of weight l. Since the technique depends considerably on this l, research efforts have been estimating the optimal value for different scenarios, including 2D grids with multiple solutions. However, specifically two previous works have used different stopping conditions for the simulations. Here, firstly, we show that these stopping conditions are not interchangeable. After doing such a pending investigation to define the stopping condition properly, we analyze the impacts of multiple solutions on the final results achieved by the technique, which is the main contribution of this work. In doing so, we demonstrate that the success probability is inversely proportional to the density of vertices marked as solutions and directly proportional to the relative distance between solutions. These relations presented here are guaranteed only for high values of the input parameters because, from different points of view, we show that a disturbed transition range exists in the small values.
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References
Aharonov, Y., Davidovich, L., Zagury, N.: Quantum random walks. Phys. Rev. A 48(2), 1687 (1993)
Ambainis, A., Bačkurs, A., Nahimovs, N., Ozols, R., Rivosh, A.: Search by quantum walks on two-dimensional grid without amplitude amplification. In: Iwama, K., Kawano, Y., Murao, M. (eds.) TQC 2012. LNCS, vol. 7582, pp. 87–97. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-35656-8_7
Ambainis, A., Kempe, J., Rivosh, A.: Coins make quantum walks faster. In: Proceedings of 16th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1099–1108. SIAM (2005)
Benioff, P.: Space searches with a quantum robot. In: Quantum Computation and Information (Washington, D.C., 2000), Contemporary Mathematics, vol. 305, pp. 1–12. American Mathematical Society, Providence (2002)
Brassard, G., Hoyer, P., Mosca, M., Tapp, A.: Quantum amplitude amplification and estimation. Contemp. Math. 305, 53–74 (2002)
Childs, A.M., Goldstone, J.: Spatial search and the Dirac equation. Phys. Rev. A 70(4), 042312 (2004)
Childs, A.M., Goldstone, J.: Spatial search by quantum walk. Phys. Rev. A 70(2), 022314 (2004)
Farhi, E., Gutmann, S.: Quantum computation and decision trees. Phys. Rev. A 58(2), 915 (1998)
Giri, P.R., Korepin, V.: Lackadaisical quantum walk for spatial search. Mod. Phys. Lett. A 35(08), 2050043 (2019)
Grover, L.K.: Quantum mechanics helps in searching for a needle in a haystack. Phys. Rev. Lett. 79(2), 325 (1997)
Høyer, P., Yu, Z.: Analysis of lackadaisical quantum walks. arXiv preprint arXiv:2002.11234 (2020)
McMahon, D.: Quantum Computing Explained. Wiley, Hoboken (2007)
Mermin, N.D.: Quantum Computer Science: An Introduction. Cambridge University Press, New York (2007)
Meyer, D.A., Wong, T.G.: Connectivity is a poor indicator of fast quantum search. Phys. Rev. Lett. 114(11), 110503 (2015)
Nahimovs, N.: Lackadaisical quantum walks with multiple marked vertices. In: Catania, B., Královič, R., Nawrocki, J., Pighizzini, G. (eds.) SOFSEM 2019. LNCS, vol. 11376, pp. 368–378. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-10801-4_29
Nahimovs, N., Rivosh, A.: Quantum walks on two-dimensional grids with multiple marked locations. Int. J. Found. Comput. Sci. 29(04), 687–700 (2018)
Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information, 10th anniversary edn. Cambridge University Press, New York (2010)
Portugal, R.: Quantum Walks and Search Algorithms. Springer, New York (2013). https://doi.org/10.1007/978-1-4614-6336-8
Portugal, R., Fernandes, T.D.: Quantum search on the two-dimensional lattice using the staggered model with Hamiltonians. Phys. Rev. A 95(4), 042341 (2017)
Rhodes, M.L., Wong, T.G.: Search on vertex-transitive graphs by lackadaisical quantum walk. arXiv preprint arXiv:2002.11227 (2020)
Saha, A., Majumdar, R., Saha, D., Chakrabarti, A., Sur-Kolay, S.: Search of clustered marked states with lackadaisical quantum walks. arXiv preprint arXiv:1804.01446 (2018)
Shenvi, N., Kempe, J., Whaley, K.B.: Quantum random-walk search algorithm. Phys. Rev. A 67(5), 052307 (2003)
Tulsi, A.: Faster quantum-walk algorithm for the two-dimensional spatial search. Phys. Rev. A 78(1), 012310 (2008)
Wong, T.G.: Grover search with lackadaisical quantum walks. J. Phys. A Math. Theor. 48(43), 435304 (2015)
Wong, T.G.: Coined quantum walks on weighted graphs. J. Phys. A Math. Theor. 50(47), 475301 (2017)
Wong, T.G.: Faster search by lackadaisical quantum walk. Quantum Inf. Process. 17(3), 1–9 (2018). https://doi.org/10.1007/s11128-018-1840-y
Yanofsky, N.S., Mannucci, M.A.: Quantum Computing for Computer Scientists. Cambridge University Press, New York (2008)
Acknowledgments
Science and Technology Support Foundation of Pernambuco (FACEPE) Brazil, Brazilian National Council for Scientific and Technological Development (CNPq), and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001 by their financial support to the development of this research.
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de Carvalho, J.H.A., de Souza, L.S., de Paula Neto, F.M., Ferreira, T.A.E. (2020). Impacts of Multiple Solutions on the Lackadaisical Quantum Walk Search Algorithm. In: Cerri, R., Prati, R.C. (eds) Intelligent Systems. BRACIS 2020. Lecture Notes in Computer Science(), vol 12319. Springer, Cham. https://doi.org/10.1007/978-3-030-61377-8_9
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