Abstract
We review the main findings on the ranking capabilities of the recently proposed Quantum PageRank algorithm (G.D. Paparo et al., Sci. Rep. 2, 444 (2012) and G.D. Paparo et al., Sci. Rep. 3, 2773 (2013)) applied to large complex networks. The algorithm has been shown to identify unambiguously the underlying topology of the network and to be capable of clearly highlighting the structure of secondary hubs of networks. Furthermore, it can resolve the degeneracy in importance of the low-lying part of the list of rankings. Examples of applications include real-world instances from the WWW, which typically display a scale-free network structure and models of hierarchical networks. The quantum algorithm has been shown to display an increased stability with respect to a variation of the damping parameter, present in the Google algorithm, and a more clearly pronounced power-law behaviour in the distribution of importance among the nodes, as compared to the classical algorithm.
Similar content being viewed by others
References
S. Brin, L. Page, Comput. Netw. ISDN Syst. 30, 107 (1998).
L. Page, S. Brin, R. Motwani, T. Winograd, The PageRank citation ranking: Bringing order to the web (1999).
A. Arratia, C. Marijuán, arXiv:1105.1595 (2011).
B. Georgeot, O. Giraud, D.L. Shepelyansky, Phys. Rev. E 81, 056109 (2010).
M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2010).
A. Galindo, M.A. Martin-Delgado, Rev. Mod. Phys. 74, 347 (2002).
L.K. Grover, A fast quantum mechanical algorithm for database search, in Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing (ACM, 1996) pp. 212--219.
A. Galindo, M.A. Martin-Delgado, Phys. Rev. A 62, 062303 (2000).
G.D. Paparo, V. Dunjko, A. Makmal, M.A. Martin-Delgado, H.J. Briegel, arXiv:1401.4997 (2014).
T.D. Ladd et al., Nature 464, 45 (2010).
G.D. Paparo, M. Martin-Delgado, Sci. Rep. 2, 444 (2012).
G.D. Paparo, M. Müller, F. Comellas, M.A. Martin-Delgado, Sci. Rep. 3, 2773 (2013).
C.H. Bennett, G. Brassard, Quantum cryptography: Public key distribution and coin tossing, in Proceedings of IEEE International Conference on Computers, Systems and Signal Processing Vol. 175 (New York, 1984) p. 8.
A.K. Ekert, Phys. Rev. Lett. 67, 661 (1991).
C. Elliott, The DARPA quantum network, in Quantum Communications and Cryptography, edited by A.V. Sergienko (CRC Press, Boca Raton, 2006) Chapt. 4, pp. 83--102.
A. Poppe, M. Peev, O. Maurhart, Int. J. Quantum Inf. 6, 209 (2008).
M. Sasaki et al., Opt. Express 19, 10387 (2011).
http://swissquantum.idquantique.com/ (17-06-2013).
D. Lancho, J. Martinez, D. Elkouss, M. Soto, V. Martin, QKD in standard optical telecommunications networks, in Quantum Communication and Quantum Networking (2010) pp. 142--149.
T. Länger, G. Lenhart, New J. Phys. 11, 055051 (2009).
H.J. Kimble, Nature 453, 1023 (2008).
D.S. Wiersma, Science 327, 1333 (2010).
M.-X. Luo, G. Xu, X.-B. Chen, Y.-X. Yang, X. Wang, Sci. Rep. 4, 4571 (2014).
H.-J. Briegel, W. Dür, J.I. Cirac, P. Zoller, Phys. Rev. Lett. 81, 5932 (1998).
W. Dür, H.J. Briegel, J. Cirac, P. Zoller, Phys. Rev. A 59, 169 (1999).
N. Sangouard, C. Simon, H. De Riedmatten, N. Gisin, Rev. Mod. Phys. 83, 33 (2011).
B. Lauritzen, J. Minář, H. De Riedmatten, M. Afzelius, N. Gisin, Phys. Rev. A 83, 012318 (2011).
C. Simon et al., Eur. Phys. J. D 58, 1 (2010).
B. Lauritzen et al., Phys. Rev. Lett. 104, 080502 (2010).
A. Acín, J.I. Cirac, M. Lewenstein, Nat. Phys. 3, 256 (2007).
S. Perseguers, J.I. Cirac, A. Acín, M. Lewenstein, J. Wehr, Phys. Rev. A 77, 022308 (2008).
Z.-W. Wei, B.-H. Wang, X.-P. Han, Sci. Rep. 3, 1222 (2013).
M. Cuquet, J. Calsamiglia, Phys. Rev. Lett. 103, 240503 (2009).
M. Cuquet, J. Calsamiglia, Phys. Rev. A 83, 032319 (2011).
D. Watts, S. Strogatz, Nature 393, 440 (1998).
A.-L. Barabási, R. Albert, Science 286, 509 (1999).
C. Song, S. Havlin, H.A. Makse, Nature 433, 392 (2005).
E. Ravasz, A.-L. Barabási, Phys. Rev. E 67, 026112 (2003).
A.-L. Barabási, Z.N. Oltvai, Nat. Rev. Genet. 5, 101 (2004).
A.-L. Barabási, Z. Dezso, E. Ravasz, S.-H. Yook, Z. Oltvai, AIP Conf. Proc. 661, 1 (2003).
M.E. Newman, SIAM Rev. 45, 167 (2003).
M. Newman, Networks: An Introduction (Oxford University Press, Inc., 2010).
R. Cohen, S. Havlin, Complex Networks: Structure, Robustness and Function (Cambridge University Press, 2010).
Timeline of web search engines --- Wikipedia, the free encyclopedia (2014), http://en.wikipedia.org/w/index.php?title=Timeline_of_web_search_engines&oldid=616361284.
M. Marchiori, Comput. Netw. ISDN Syst. 29, 1225 (1997).
A.N. Langville, C.D. Meyer, Internet Math. 1, 335 (2004).
C.D. Meyer, Matrix Analysis and Applied Linear Algebra, Vol. 2 (Siam, 2000).
T. Haveliwala, S. Kamvar, The second eigenvalue of the Google matrix, Stanford University Technical Report (2003).
M. Szegedy, Quantum speed-up of Markov chain based algorithms, in Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, 2004 (2004) pp. 32--41.
D. Aharonov, A. Ambainis, J. Kempe, U. Vazirani, Quantum walks on graphs, in Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing (ACM, 2001) pp. 50--59.
R. Albert, A.-L. Barabási, Rev. Mod. Phys. 74, 47 (2002).
S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, D.-U. Hwang, Phys. Rep. 424, 175 (2006).
A.-L. Barabási, R. Albert, H. Jeong, Physica A: Stat. Mech. Appl. 281, 69 (2000).
A. Barrat, M. Barthelemy, R. Pastor-Satorras, A. Vespignani, Proc. Natl. Acad. Sci. U.S.A. 101, 3747 (2004).
H. Jeong, P. Mason, A.-L. Barabasi, Z. Oltvai, Nature 411, 41 (2001).
H. Jeong, B. Tombor, R. Albert, Z.N. Oltvai, A.-L. Barabási, Nature 407, 651 (2000).
R. Albert, H. Jeong, A.-L. Barabási, Nature 406, 378 (2000).
D.S. Callaway, M.E. Newman, S.H. Strogatz, D.J. Watts, Phys. Rev. Lett. 85, 5468 (2000).
A. Vázquez, Y. Moreno, Phys. Rev. E 67, 015101 (2003).
M. Boguna, D. Krioukov, K. Claffy, Nat. Phys. 5, 74 (2008).
S. Carmi, S. Carter, J. Sun, D. Ben-Avraham, Phys. Rev. Lett. 102, 238702 (2009).
S.H. Lee, P. Holme, Phys. Rev. Lett. 108, 128701 (2012).
Y.-Y. Liu, J.-J. Slotine, A.-L. Barabási, Nature 473, 167 (2011).
T. Nepusz, T. Vicsek, Nat. Phys. 8, 568 (2012).
V. Nicosia, R. Criado, M. Romance, G. Russo, V. Latora, Sci. Rep. 2, 218 (2012).
A.-L. Barabási, Philos. Trans. R. Soc. A 371, 20120512 (2013).
A. Reka, H. Jeong, A.-L. Barabasi, Nature 401, 130 (1999).
B. Bollobás, C. Borgs, J. Chayes, O. Riordan, Directed scale-free graphs, in Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2003) pp. 132--139.
A. Hagberg, P. Swart, D.S. Chult, Exploring network structure, dynamics, and function using NetworkX Tech. Rep., Los Alamos National Laboratory (LANL) (2008).
V. Batagelj, A. Mrvar, Pajek datasets, http://vlado.fmf.uni-lj.si/pub/networks/data (2006).
E. Ravasz, A.L. Somera, D.A. Mongru, Z.N. Oltvai, A.-L. Barabási, Science 297, 1551 (2002).
A.-L. Barabási, E. Ravasz, T. Vicsek, Physica A: Stat. Mech. Appl. 299, 559 (2001).
J.D. Noh, Phys. Rev. E 67, 045103 (2003).
F. Comellas, A. Miralles, Physica A: Stat. Mech. Appl. 388, 2227 (2009).
F. Comellas, A. Miralles, J. Phys. A: Math. Theor. 42, 425001 (2009).
H.K. Lo, T. Spiller, S. Popescu, Introduction to Quantum Computation and Information (World Scientific Publishing Company, 1998).
D. Donato, L. Laura, S. Leonardi, S. Millozzi, Eur. Phys. J. B 38, 239 (2004).
G. Pandurangan, P. Raghavan, E. Upfal, Using PageRank to Characterize Web Structure, in Computing and Combinatorics, Vol. 2387 (Springer, 2002) pp. 330--339.
F. de Lima Marquezino, R. Portugal, S. Boettcher, Phys. Rev. A 87, 012329 (2013).
F. Caruso, arXiv:1312.1832 (2013).
A. Makmal, M. Zhu, D. Manzano, M. Tiersch, H.J. Briegel, arXiv:1309.5253 (2013).
S.E. Venegas-Andraca, Quantum Inf. Process. 11, 1015 (2012).
M. Faccin, T. Johnson, J. Biamonte, S. Kais, P. Migdał, Phys. Rev. X 3, 041007 (2013).
B. Kollár, J. Novotny, T. Kiss, I. Jex, arXiv:1404.4509 (2014).
J.D. Whitfield, Reflections in Hilbert Space II: Szegedy's Scheme for Markov Chain Quantization (2012).
S. Garnerone, P. Zanardi, D.A. Lidar, Phys. Rev. Lett. 108, 230506 (2012).
S. Garnerone, Phys. Rev. A 86, 032342 (2012).
E. Sánchez-Burillo, J. Duch, J. Gómez-Gardeñes, D. Zueco, Sci. Rep. 2, 605 (2012).
F. Verstraete, M.M. Wolf, J.I. Cirac, Nat. Phys. 5, 633 (2009).
S. Diehl et al., Nat. Phys. 4, 878 (2008).
A.M. Childs, D. Gosset, Z. Webb, Science 339, 791 (2013).
A.M. Childs, Phys. Rev. Lett. 102, 180501 (2009).
Author information
Authors and Affiliations
Corresponding author
Additional information
Contribution to the Focus Point “Quantum information and complexity” edited by S. Mancini, G. Marmo, S. Pascazio.
This contribution is based on the talk presented by G.D. Paparo at the workshop Noise Information & Complexity at Quantum Scale at the Ettore Majorana Centre, Erice (Sicily), Italy 6th–12th October 2013.
Rights and permissions
About this article
Cite this article
Paparo, G.D., Müller, M., Comellas, F. et al. Quantum Google algorithm. Eur. Phys. J. Plus 129, 150 (2014). https://doi.org/10.1140/epjp/i2014-14150-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/i2014-14150-y