Abstract
Discrete quantum walks are dynamical protocols for controlling a single quantum particle. Despite of its simplicity, quantum walks display rich topological phenomena and provide one of the simplest systems to study and understand topological phases. In this article, we review the physics of discrete quantum walks in one and two dimensions in light of topological phenomena and provide elementary explanations of topological phases and their physical consequence, namely the existence of boundary states. We demonstrate that quantum walks are versatile systems that simulate many topological phases whose classifications are known for static Hamiltonians. Furthermore, topological phenomena appearing in quantum walks go beyond what has been known in static systems; there are phenomena unique to quantum walks, being an example of periodically driven systems, that do not exist in static systems. Thus the quantum walks not only provide a powerful tool as a quantum simulator for static topological phases but also give unique opportunity to study topological phenomena in driven systems.
Similar content being viewed by others
References
Aharonov Y., Davidovich L., Zagury N.: Quantum random walks. Phys. Rev. A 48, 1687 (1993)
Kempe, J.: Quantum random walks: an introductory overview. Contemp. Phys. 44, 307 (2003). http://www.tandfonline.com/doi/pdf/10.1080/00107151031000110776
Farhi E., Gutmann S.: Quantum computation and decision trees. Phys. Rev. A 58, 915 (1998)
Kitagawa T., Rudner M.S., Berg E., Demler E.: Exploring topological phases with quantum walks. Phys. Rev. A 82, 033429 (2010)
Kitagawa, T., et al.: Observation of topologically protected bound states in a one dimensional photonic system. arXiv/1105.5334 (2011)
Zähringer F.: Realization of a quantum walk with one and two trapped ions. Phys. Rev. Lett. 104, 100503 (2010)
Karski, M., et al.: Quantum walk in position space with single optically trapped atoms. Science 325, 174 (2009). http://www.sciencemag.org/content/325/5937/174.full.pdf
Schreiber A., et al.: Photons walking the line: a quantum walk with adjustable coin operations. Phys. Rev. Lett. 104, 050502 (2010)
Broome M.A. et al.: Discrete single-photon quantum walks with tunable decoherence. Phys. Rev. Lett. 104, 153602 (2010)
Biane, P., et al.: Quantum walks. In: Quantum Potential Theory, Lecture Notes in Mathematics, vol. 1954, pp. 309–452. Springer, Berlin, Heidelberg (2008)
Grimmett G., Janson S., Scudo P.F.: Weak limits for quantum random walks. Phys. Rev. E 69, 026119 (2004)
Ryu S., Hatsugai Y.: Topological origin of zero-energy edge states in particle-hole symmetric systems. Phys. Rev. Lett. 89, 077002 (2002)
Oka T., Konno N., Arita R., Aoki H.: Breakdown of an electric-field driven system: a mapping to a quantum walk. Phys. Rev. Lett. 94, 100602 (2005)
Rudner M.S., Levitov L.S.: Topological transition in a non-hermitian quantum walk. Phys. Rev. Lett. 102, 065703 (2009)
von Klitzing K., Dorda G., Pepper M.: New method for high-accuracy determination of the finestructure constant based on quantized hall resistance. Phys. Rev. Lett. 45, 494 (1980)
MacDonald, A.H.: Introduction to the physics of the quantum hall regime: quantized hall conductance in a two-dimensional periodic potential. eprint arXiv:cond-mat/9410047 (1994)
Thouless D.J., Kohmoto M., Nightingale M.P., den Nijs M.: Quantized hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405 (1982)
Halperin B.I.: Quantized hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential. Phys. Rev. B 25, 2185 (1982)
Hasan M.Z., Kane C.L.: Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045 (2010)
Qi X.-L., Zhang S.-C.: Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057 (2011)
Su W.P., Schrieffer J.R., Heeger A.J.: Solitons in polyacetylene. Phys. Rev. Lett. 42, 1698 (1979)
Jackiw R., Rebbi C.: Solitons with fermion number. Phys. Rev. D 13, 3398 (1976)
Jackiw R., Schrieffer J.: Solitons with fermion number 12 in condensed matter and relativistic field theories. Nuclear Physics B 190, 253 (1981)
Kane C.L., Mele E.J.: Quantum spin hall effect in graphene. Phys. Rev. Lett. 95, 226801 (2005)
Bernevig, B.A., Hughes, T.L., Zhang, S.-C.: Quantum spin hall effect and topological phase transition in HgTe quantum wells. Science 314, 1757 (2006). http://www.sciencemag.org/content/314/5806/1757.full.pdf
Konig, M., et al.: Quantum spin hall insulator state in HgTe quantum wells. Science 318, 766 (2007). http://www.sciencemag.org/content/318/5851/766.full.pdf
Fu L., Kane C.L., Mele E.J.: Topological insulators in three dimensions. Phys. Rev. Lett. 98, 106803 (2007)
Chen, Y.L., et al.: Experimental realization of a three-dimensional topological insulator, Bi2Te3. Science 325, 178 (2009). http://www.sciencemag.org/content/325/5937/178.full.pdf
Xia Y. et al.: Observation of a large-gap topological-insulator class with a single Dirac cone on the surface. Nat. Phys. 5, 398 (2009)
Schnyder A.P., Ryu S., Furusaki A., Ludwig A.W.W.: Classification of topological insulators and superconductors in three spatial dimensions. Phys. Rev. B 78, 195125 (2008)
Qi X.-L., Hughes T.L., Zhang S.-C.: Topological field theory of time-reversal invariant insulators. Phys. Rev. B 78, 195424 (2008)
Kitaev, A.: Periodic table for topological insulators and superconductors. In: Lebedev V., Feigel’Man, M. (eds.) American Institute of Physics Conference Series, vol. 1134, pp. 22–30 (2009)
Wang Z., Chong Y., Joannopoulos J.D., Soljacic M.: Observation of unidirectional backscattering–immune topological electromagnetic states. Nature 461, 772 (2009)
Kitagawa T., Berg E., Rudner M., Demler E.: Topological characterization of periodically driven quantum systems. Phys. Rev. B 82, 235114 (2010)
Jiang L. et al.: Majorana Fermions in equilibrium and in driven cold-atom quantum wires. Phys. Rev. Lett. 106, 220402 (2011)
Sørensen A.S., Demler E., Lukin M.D.: Fractional quantum hall states of atoms in optical lattices. Phys. Rev. Lett. 94, 086803 (2005)
Zhu S.-L., Fu H., Wu C.-J., Zhang S.-C., Duan L.-M.: Spin hall effects for cold atoms in a light-induced gauge potential. Phys. Rev. Lett. 97, 240401 (2006)
Jaksch D., Zoller P.: Creation of effective magnetic fields in optical lattices: the Hofstadter butterfly for cold neutral atoms. N. J. Phys. 5, 56 (2003)
Osterloh K., Baig M., Santos L., Zoller P., Lewenstein M.: Cold atoms in non-abelian gauge potentials: from the hofstadter “Moth” to lattice gauge theory. Phys. Rev. Lett. 95, 010403 (2005)
Kitagawa, T.: Wolfram Demonstration: topological phases with quantum walks. http://demonstrations.wolfram.com/TopologicalPhasesWithQuantumWalks/
Thouless D.J.: Quantization of particle transport. Phys. Rev. B 27, 6083 (1983)
Obuse H., Kawakami N.: Topological phases and delocalization of quantum walks in random environments. Phys. Rev. B 84, 195139 (2011)
Kitagawa, T., Oka, T., Demler, E.: Photo control of transport properties in disorderd wire; average conductance, conductance statistics, and time-reversal symmetry. WolframDemonstration: topological phases with quantum walks. ArXiv e-prints (2012). 1201.0521
Moore J.E., Ran Y., Wen X.-G.: Topological surface states in three-dimensional magnetic insulators. Phys. Rev. Lett. 101, 186805 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kitagawa, T. Topological phenomena in quantum walks: elementary introduction to the physics of topological phases. Quantum Inf Process 11, 1107–1148 (2012). https://doi.org/10.1007/s11128-012-0425-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11128-012-0425-4