Abstract
The quantum walk (QW) is the term given to a family of algorithms governing the evolution of a discrete quantum system and as such has a founding role in the study of quantum computation. We contribute to the investigation of QW phenomena by performing a detailed numerical study of discrete-time quantum walks. In one dimension (1D), we compute the structure of the probability distribution, which is not a smooth curve but shows oscillatory features on all length scales. By analyzing walks up to N = 1,000,000 steps, we discuss the scaling characteristics and limiting forms of the QW in both real and Fourier space. In 2D, with a view to ready experimental realization, we consider two types of QW, one based on a four-faced coin and the other on sequential flipping of a single two-faced coin. Both QWs may be generated using two two-faced coins, which in the first case are completely unentangled and in the second are maximally entangled. We draw on our 1D results to characterize the properties of both walks, demonstrating maximal speed-up and emerging semi-classical behavior in the maximally entangled QW.
Similar content being viewed by others
References
Kempe, J.: Quantum random walks: an introductory overview. Contemp. Phys. 44, 307–327 (2003)
Aharonov, Y., Davidovich, L., Zagury, N.: Quantum random walks. Phys. Rev. A 48, 1687–1690 (1993)
Shenvi, N., Kempe, J., Whaley, K.B.: Quantum random-walk search algorithm. Phys. Rev. A 67, 052307 (2003)
Ambainis, A.: Quantum walks and their algorithmic applications. Int. J. Quantum Inf. 1, 507 (2003)
Kendon, V.: A random walk approach to quantum algorithms. Phil. Trans. R. Soc. A 364, 3407 (2006)
Childs, A.M., Cleve, R., Deotto, E., Farhi, E., Gutmann, S., Spielman, D.A.: Exponential algorithmic speedup by a quantum walk. In: Proceedings of the 35th Annual ACM Symposium on Theory of Computing, STOC 03, pp. 59–68 (2003)
Kempe, J.: Discrete quantum walks hit exponentially faster. In: Proceedings of the 7th International Workshop on Randomization and Approximation Techniques in Computer Science (RANDOM03), pp. 354–369 (2003)
Ambainis, A., Kempe, J., Rivosh, A.: Coins make quantum walks faster. In: Proceedings of 16th ACM-SIAM SODA, pp. 1099–1108 (2005)
Ambainis, A., Bach, E., Nayak, A., Vishwanath, A., Watrous, J.: One-dimensional quantum walks. In: Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, STOC, vol. 01, pp. 37–49 (2001)
Childs, A.M., Farhi, E., Gutmann, S.: An example of the difference between quantum and classical random walks. Quantum Inf. Process. 1, 35 (2002)
Tregenna, B., Flanagan, W., Maile, R., Kendon, V.: Controlling discrete quantum walks: coins and initial states. New J. Phys. 5, 83 (2003)
Brun, T.A., Carteret, H.A., Ambainis, A.: Quantum walks driven by many coins. Phys. Rev. A 67, 052317 (2003)
Xue, P., Sanders, B.C.: Two quantum walkers sharing coins. Phys. Rev. A 85, 022307 (2012)
Xue, P.: Implementation of multi-walker quantum walks with cavity grid. J. Comput. Theor. Nanosci. 10, 1606–1612 (2013)
Mackay, T.D., Bartlett, S.D., Stephenson, L.T., Sanders, B.C.: Quantum walks in higher dimensions. J. Phys. A Math. Gen. 35, 2745 (2002)
Zhang, R., Xue, P.: Two-dimensional quantum walk with position-dependent phase defects. Quantum Inf. Process. 13, 1825–1839 (2014)
Di Franco, C., McGettrick, M., Machida, T., Busch, T.: Alternate two-dimensional quantum walk with a single-qubit coin. Phys. Rev. A 84, 042337 (2011)
Di Franco, C., McGettrick, M., Busch, T.: Mimicking the probability distribution of a two-dimensional grover walk with a single-qubit coin. Phys. Rev. Lett. 106, 080502 (2011)
Do, B., Stohler, M.L., Balasubramanian, S., Elliott, D.S., Eash, C., Fischbach, E., Fischbach, M.A., Mills, A., Zwickl, B.: Experimental realization of a quantum quincunx by use of linear optical elements. J. Opt. Soc. Am. B 22, 499–504 (2005)
Schreiber, A., Cassemiro, K.N., Potoc̆ek, V., G’abris, A., Mosley, P.J., Andersson, E., Jex, I., Silberhorn, C.: Photons walking the line: a quantum walk with adjustable coin operations. Phys. Rev. Lett. 104, 050502 (2010)
Zhang, P., Ren, X.F., Zou, X.B., Liu, B.H., Huang, Y.F., Guo, G.C.: Demonstration of one-dimensional quantum random walks using orbital angular momentum of photons. Phys. Rev. A 75, 052310 (2007)
Zhang, P., Liu, B.H., Liu, R.F., Li, H.R., Li, F.L., Guo, G.C.: Implementation of one-dimensional quantum walks on spin-orbital angular momentum space of photons. Phys. Rev. A 81, 052322 (2010)
Karski, M., Förster, L., Choi, J.M., Steffen, A., Alt, W., Meschede, D., Widera, A.: Quantum walk in position space with single optically trapped atoms. Science 325, 174–177 (2009)
Schmitz, H., Matjeschk, R., Schneider, C., Glueckert, J., Enderlein, M., Huber, T., Schaetz, T.: Quantum walk of a trapped ion in phase space. Phys. Rev. Lett. 103, 090504 (2009)
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)
Ryan, C.A., Laforest, M., Boileau, J.C., Laflamme, R.: Experimental implementation of a discrete–time quantum random walk on an nmr quantum-information processor. Phys. Rev. A 72, 062317 (2005)
Perets, H.B., Lahini, Y., Pozzi, F., Sorel, M., Morandotti, R., Silberberg, Y.: Realization of quantum walks with negligible decoherence in waveguide lattices. Phys. Rev. Lett. 100, 170506 (2008)
Schreiber, A., Gábris, A., Rohde, P.P., Laiho, K., Stefanak, M., Potoc̈ek, V., Hamilton, C., Jex, I., Silberhorn, C.: A 2d quantum walk simulation of two-particle dynamics. Science 336, 55–58 (2012)
Xue, P., Zhang, R., Qin, H., Zhan, X., Bian, Z.H., Li, J., Sanders, B.C.: Experimental quantum-walk revival with a time-dependent coin. Phys. Rev. Lett. 114, 140502 (2015)
Bian, Z., Li, J., Qin, H., Zhan, X., Zhang, R., Sanders, B.C., Xue, P.: Realization of single-qubit positive-operator-valued measurement via a one-dimensional photonic quantum walk. Phys. Rev. Lett. 114, 203602 (2015)
Xue, P., Qin, H., Tang, B., Sanders, B.C.: Observation of quasiperiodic dynamics in a one-dimensional quantum walk of single photons in space. New J. Phys. 16, 053009 (2014)
Xue, P., Qin, H., Tang, B., Sanders, B.C.: Trapping photons on the line: controllable dynamics of a quantum walk. Sci. Rep. 4, 04825 (2014)
Marquezino, F., Portugal, R.: The QWalk simulator of quantum walks. Comput. Phys. Commun. 179, 359–369 (2008)
Sawerwain, M., Gielerak, R.: GPGPU based simulations for one and two dimensional quantum walks. In: Kwiecień, A., Gaj, P., Stera, P. (eds.) Computer Networks, pp. 29–38. Springer, Berlin (2010)
Konno, N.: A new type of limit theorems for the one-dimensional quantum random walk. J. Math. Soc. Jpn. 57, 1179 (2005)
Grimmett, G., Janson, S., Scudo, P.F.: Weak limits for quantum random walks. Phys. Rev. E 69, 026119 (2004)
Oliveira, A.C., Portugal, R., Donangelo, R.: Decoherence in two-dimensional quantum walks. Phys. Rev. A 74, 012312 (2006)
Inui, N., Konishi, Y., Konno, N.: Localization of two-dimensional quantum walks. Phys. Rev. A 69, 052323 (2004)
Roldán, E., Di Franco, C., Silva, F., de Valcárcel, G.J.: N-dimensional alternate coined quantum walks from a dispersion-relation perspective. Phys. Rev. A 87, 022336 (2013)
Acknowledgments
The authors gratefully acknowledge helpful discussions with Professor B. Normand. Work at Renmin University was supported by the National Natural Science Foundation of China (NSFC) under Grant No. 11174365 and by the National Basic Research Program of China (NBRPC) under Grant No. 2012CB921704. PX was supported by the NSFC under Grant Nos. 11174052 and 11474049, by the NBRPC under Grant No. 2011CB921203, by the Open Fund from the State Key Laboratory of Precision Spectroscopy of East China Normal University, and by the CAST Innovation Fund.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Luo, H., Xue, P. Properties of long quantum walks in one and two dimensions. Quantum Inf Process 14, 4361–4394 (2015). https://doi.org/10.1007/s11128-015-1127-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11128-015-1127-5