A remark on zeta functions of finite graphs via quantum walks

Ambainis, A.: Quantum walks and their algorithmic applications. Int. J. Quantum Inf. 1, 507–518 (2003).

Ambainis, A., Bach, E., Nayak, A., Vishwanath, A., Watrous, J.: One-dimensional quantum walks. In: Proc. 33rd Annual ACM Symp. Theory of Computing, pp. 37–49 (2001).

Bass, H.: The Ihara-Selberg zeta function of a tree lattice. Internat. J. Math. 3, 717–797 (1992).

Biggs, N.: Algebraic Graph Theory. Cambridge Univ. Press, Cambridge, UK (1974).

Chandrashekar, C.M., Banerjee, S., Srikanthm, R.: Relationship between quantum walk and relativistic quantum mechanics. Phys. Rev. A. 81, 062340 (2010).

Childs, A.M.: Universal computation by quantum walk. Phys. Rev. Lett. 102, 180501 (2009).

Cvetković, D.M., Doob, M., Sachs, H.: Spectra of Graphs. Academic Press, New York (1979).

Cvetković, D., Rowlinson, P., Simić, S.K.: Signless Laplacians of finite graphs. Linear Algebra Appl. 423, 155–171 (2007).

Emms, D., Hancock, E.R., Severini, S., Wilson, R.C.: A matrix representation of graphs and its spectrum as a graph invariant. Electr. J. Combin. 13, R34 (2006).

Emms, D., Severini, S., Wilson, R.C., Hancock, E.R.: Coined quantum walks lift the cospectrality of graphs and trees. Pattern Recognit. 42, 1988–2002 (2009).

Gamble, J.K., Friesen, M., Zhou, D., Joynt, R., Coppersmith, S.N.: Two particle quantum walks applied to the graph isomorphism problem. Phys. Rev. A. 81, 52313 (2010).

Gantmacher, F.R.: Theory of Matrices, 2 vol. Chelsea Publishing Co, Chelsea (1959).

Godsil, C., Guo, K.: Quantum walks on regular graphs and eigenvalues. Electron. J. Combin. 18, P165 (2011).

Godsil, C., Royle, G.: Algebraic Graph Theory. Springer-Verlag, New York (2001).

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

Grover, L.K.: From Schrödinger’s equation to quantum search algorithm. Am. J. Phys. 69, 769–777 (2001).

Hashimoto, K.: Zeta functions of finite graphs and representations of p-Adic groups. Adv. Stud. Pure Math. 15, 211–280 (1989).

Higuchi, Yu., Konno, N., Sato, I., Segawa, E.: A note on the discrete-time evolutions of quantum walk on a graph. J. Math-for-Ind. 5B, 103–109 (2013).

Ihara, Y.: On discrete subgroups of the two by two projective linear group over p-adic fields. J. Math. Soc. Japan. 18, 219–235 (1966).

Karski, M., Föster, L., Choi, J-M., Steffen, A., Alt, W., Meschede, D., Widera, A.: Quantum walk in position space with single optically trapped atoms. Science. 325, 174 (2009).

Kempe, J.: Quantum random walks - an introductory overview. Contemporary Phys. 44, 307–327 (2003).

Konno, N.: Quantum random walks in one dimension. Quantum Inf. Process. 1, 345–354 (2002).

Konno, N.: A new type of limit theorems for the one-dimensional quantum random walk. J. Math. Soc. Japan. 57, 1179–1195 (2005).

Konno, N.: Quantum walks. Lect. Notes Math. 1954, 309–452 (2008).

Konno, N., Sato, I.: On the relation between quantum walks and zeta functions. Quantum Inf. Process. 11, 341–349 (2012).

Kotani, M., Sunada, T.: Zeta functions of finite graphs. J. Math. Sci. U. Tokyo. 7, 7–25 (2000).

Magniez, F., Nayak, A., Roland, J., Santha, M.: Search via quantum walk. In: Proc. 39th ACM Symposium on Theory of Computing, pp. 575–584 (2007).

Manouchehri, K., Wang, J.: Physical Implementation of Quantum Walks, Quantum Science and Technology. Springer, Berlin Heidelberg (2014).

Matsuoka, L., Yokoyama, K.: Physical implementation of quantum cellular automaton in a diatomic molecule. Special issue: “Theoretical and mathematical aspects of the discrete time quantum walk”. J. Comput. Theor. Nanosci. 10, 1617–1620 (2013).

Mohseni, M., Rebentrost, P., Lloyd, S., Aspuru-Guzik, A.: Environment-assisted quantum walks in photosynthetic energy transfer. J. Chem. Phys. 129, 174106 (2008).

Northshield, S.: A note on the zeta function of a graph. J. Combin. Theory Ser. B. 74, 408–410 (1998).

Ren, P., Aleksic, T., Emms, D., Wilson, R.C., Hancock, E.R.: Quantum walks, Ihara zeta functions and cospectrality in regular graphs. Quantum Inf. Proc. 10, 405–417 (2011).

Schreiber, A., Cassemiro, K.N., Potoček, V., Gábris, A., Mosley, P.J., Anderson, E., Jex, I., Silberhorn, Ch.: Photons walking the line: a quantum walk with adjustable coin operations. Phys. Rev. Lett. 104, 050502 (2010).

Segawa, E.: Localization of quantum walks induced by recurrence properties of random walks. Special issue: “Theoretical and mathematical aspects of the discrete time quantum walk”. J. Comput. Theor. Nanosci. 10, 1583–1590 (2013).

Shiau, S-Y., Joynt, R., Coppersmith, S.N.: Physically-motivated dynamical algorithms for the graph isomorphism problem. Quantum Inform. Comput. 5, 492–506 (2005).

Smilansky, U.: Quantum chaos on discrete graphs. J. Phys. A: Math. Theor. 40, F621–F630 (2007).

Strauch, F.W.: Connecting the discrete- and continuous-time quantum walks. Phys. Rev. A. 74, 030301 (2006).

Sunada, T.: L-Functions in geometry and some applications. Lect. Notes Math. 1201, 266–284 (1986).

Szegedy, M.: Quantum speed-up of Markov chain based algorithms. In: Proc. 45th IEEE Symposium on Foundations of Computer Science, pp. 32–41 (2004).

Venegas-Andraca, S.E.: Quantum walks: a comprehensive review. Quantum Inf. Process. 11, 1015–1106 (2012).

Zähringer, F., Kirchmair, G., Gerritsma, R., Solano, E., Blatt, R., Roos, C.F.: Realization of a quantum walk with one and two trapped ions. Phys. Rev. Lett. 104, 100503 (2010).