|
|
|
Stopping time of a one-dimensional bounded quantum walk |
| Hao Luo(骆浩)1, Xiang Zhan(詹翔)2, Peng Zhang(张芃)1, Peng Xue(薛鹏)2 |
1 Shandong Provincial Key Laboratory of Laser Polarization and Information Technology, Department of Physics, Qufu Normal University, Qufu 273165, China;
2 Department of Physics and Information Engineering, Jining University, Qufu 273155, China |
|
|
|
|
Abstract The stopping time of a one-dimensional bounded classical random walk (RW) is defined as the number of steps taken by a random walker to arrive at a fixed boundary for the first time. A quantum walk (QW) is a non-trivial generalization of RW, and has attracted a great deal of interest from researchers working in quantum physics and quantum information. In this paper, we develop a method to calculate the stopping time for a one-dimensional QW. Using our method, we further compare the properties of stopping time for QW and RW. We find that the mean value of the stopping time is the same for both of these problems. However, for short times, the probability for a walker performing a QW to arrive at the boundary is larger than that for a RW. This means that, although the mean stopping time of a quantum and classical walker are the same, the quantum walker has a greater probability of arriving at the boundary earlier than the classical walker.
|
Received: 22 June 2016
Revised: 29 July 2016
Accepted manuscript online:
|
|
PACS:
|
03.67.Ac
|
(Quantum algorithms, protocols, and simulations)
|
| |
05.40.Fb
|
(Random walks and Levy flights)
|
|
| Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 11222430, 11434011, and 11474049), the National Basic Research Program of China (Grant No. 2012CB922104), the Fundamental Research Funds for the Central Universities, China, and the Research Funds of Renmin University of China (Grant No. 16XNLQ03). |
Corresponding Authors:
Peng Xue
E-mail: gnep.eux@gmail.com
|
Cite this article:
Hao Luo(骆浩), Xiang Zhan(詹翔), Peng Zhang(张芃), Peng Xue(薛鹏) Stopping time of a one-dimensional bounded quantum walk 2016 Chin. Phys. B 25 110304
|
| [1] |
Pearson K 1905 Nature 72 294
|
| [2] |
Aharonov Y, Davidovich L and Zagury N 1993 Phys. Rev. A 48 1687
|
| [3] |
Kempe J 2003 Contemp. Phys. 44 307
|
| [4] |
Shenvi N, Kempe J and Whaley K B 2003 Phys. Rev. A 67 052307
|
| [5] |
Ambainis A 2003 Int. J. Quantum Inf. 1 507
|
| [6] |
Kendon V 2006 Phil. Trans. R. Soc. A 364 3407
|
| [7] |
Childs A M, Cleve R, Deotto E, Farhi E, Gutmann S and Spielman D A 2003 Proceedings of the thirty-fifth Annual ACM Symposium on Theory of Computing (STOC'03), New York, USA, p. 354
|
| [8] |
Kempe J 2003 Proceedings of the 7th International Workshop on Randomization and Approximation Techniques in Computer Science (RANDOM'03), Princeton, New York, USA, p. 354
|
| [9] |
Ambainis A, Kempe J and Rivosh A 2005 Proceedings of 16th ACM-SIAM Symposium on Discrete Algorithms (SODA'05), January 23-25, 2005, Vancouver, BC, Canada, p. 1099
|
| [10] |
Zhang Y C, Bao W S, Wang X and Fu X Q 2015 Chin. Phys. B 24 110309
|
| [11] |
Ambainis A, Bach E, Nayak A, Vishwanath A and Watrous J 2001 Proceedings of the thirty-third Annual ACM Symposium on Theory of Computing (STOC'01), New York, USA, p. 37
|
| [12] |
Childs A M, Farhi E and Gutmann S 2002 Quantum Inf. Process. 1 35
|
| [13] |
Tregenna B, Flanagan W, Maile R and Kendon V 2003 New J. Phys. 5 83
|
| [14] |
Brun T A, Carteret H A and Ambainis A 2003 Phys. Rev. A 67 052317
|
| [15] |
Xue P and Sanders B C 2012 Phys. Rev. A 85 022307
|
| [16] |
Xue P 2013 J. Comput. Theor. Nanosci. 10 1606
|
| [17] |
Mackay T D, Bartlett S D, Stephenson L T and Sanders B C 2002 J. Phys. A:Math. Gen. 35 2745
|
| [18] |
Zhang R and Xue P 2014 Quantum Inf. Process. 13 1825
|
| [19] |
Di Franco C, McGettrick M, Machida T and Busch T 2011 Phys. Rev. A 84 042337
|
| [20] |
Di Franco C, McGettrick M and Busch Th 2011 Phys. Rev. Lett. 106 080502
|
| [21] |
Zhang R, Xu Y Q and Xue P 2015 Chin. Phys. B 24 010303
|
| [22] |
Qin H and Xue P 2016 Chin. Phys. B 25 010501
|
| [23] |
Li M, Zhang Y S and Guo G C 2013 Chin. Phys. B 22 030310
|
| [24] |
Zhang R, Qin H, Tang B and Xue P 2013 Chin. Phys. B 22 110312
|
| [25] |
Do B, Stohler M L, Balasubramanian S, Elliott D S, Eash C, Fischbach E, Fischbach M A, Mills A and Zwickl B 2005 J. Opt. Soc. Am. B 22 499
|
| [26] |
Schreiber A, Cassemiro K N, Potoček V, Gábris A, Mosley P J, Andersson E, Jex I and Silberhorn C 2010 Phys. Rev. Lett. 104 050502
|
| [27] |
Zhang P, Ren X F, Zou X B, Liu B H, Huang Y F and Guo G C 2007 Phys. Rev. A 75 052310
|
| [28] |
Zhang P, Liu B H, Liu R F, Li H R, Li F L and Guo G C 2010 Phys. Rev. A 81 052322
|
| [29] |
Xue P, Qin H, Tang B, Zhan X, Bian Z H and Li J 2014 Chin. Phys. B 23 110307
|
| [30] |
Bian Z H, Li J, Qin H, Zhan X, Zhang R, Sanders B C and Xue P 2015 Phys. Rev. Lett. 114 203602
|
| [31] |
Xue P, Zhang R, Qin H, Zhan X, Bian Z H, Li J and Sanders B C 2015 Phys. Rev. Lett. 114 140502
|
| [32] |
Karski M, Förster L, Choi J M, Steffen A, Alt W, Meschede D and Widera A 2009 Science 325 174
|
| [33] |
Schmitz H, Matjeschk R, Schneider C, Glueckert J, Enderlein M, Huber T and Schaetz T 2009 Phys. Rev. Lett. 103 090504
|
| [34] |
Zähringer F, Kirchmair G, Gerritsma R, Solano E, Blatt R and Roos C F 2010 Phys. Rev. Lett. 104 100503
|
| [35] |
Du J F, Li H, Xu X D, Shi M J, Wu J H, Zhou X Y and Han R D 2003 Phys. Rev. A 67 042316
|
| [36] |
Ryan C A, Laforest M, Boileau J C and Laflamme R 2005 Phys. Rev. A 72 062317
|
| [37] |
Perets H B, Lahini Y, Pozzi F, Sorel M, Morandotti R and Silberberg Y 2008 Phys. Rev. Lett. 100 170506
|
| [38] |
Schreiber A, Gábris A, Rohde P P, Laiho K, Stefanak M, Potoček V, Hamilton C, Jex I and Silberhorn C 2012 Science 336 55
|
| [39] |
Qin H and Xue P 2014 Chin. Phys. B 23 010301
|
| [40] |
Bian Z H, Qin H, Zhan X, Li J and Xue P 2016 Chin. Phys. B 25 020307
|
| [41] |
Norio K 2005 J. Math. Soc. Jpn. 57 1179
|
| [42] |
Shiryaev A N 2016 Probability-1 (3rd edn.) (New York:Springer-Verlag)
|
| [43] |
Luo H and Xue P 2015 Quantum Inf. Process. 14 4361
|
| [44] |
Brun T A, Carteret H A and Ambainis A 2003 Phys. Rev. Lett. 91 130602
|
| No Suggested Reading articles found! |
|
|
Viewed |
|
|
|
Full text
|
|
|
|
|
Abstract
|
|
|
|
|
Cited |
|
|
|
|
Altmetric
|
|
blogs
Facebook pages
Wikipedia page
Google+ users
|
Online attention
Altmetric calculates a score based on the online attention an article receives. Each coloured thread in the circle represents a different type of online attention. The number in the centre is the Altmetric score. Social media and mainstream news media are the main sources that calculate the score. Reference managers such as Mendeley are also tracked but do not contribute to the score. Older articles often score higher because they have had more time to get noticed. To account for this, Altmetric has included the context data for other articles of a similar age.
View more on Altmetrics
|
|
|
