Please wait a minute...
Chin. Phys. B, 2016, Vol. 25(11): 110304    DOI: 10.1088/1674-1056/25/11/110304
GENERAL Prev   Next  

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.

Keywords:  quantum walk      stopping time      cumulative probability  
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
[1] Quantum search of many vertices on the joined complete graph
Tingting Ji(冀婷婷), Naiqiao Pan(潘乃桥), Tian Chen(陈天), and Xiangdong Zhang(张向东). Chin. Phys. B, 2022, 31(7): 070504.
[2] Efficient quantum private comparison protocol based on one direction discrete quantum walks on the circle
Jv-Jie Wang(王莒杰), Zhao Dou(窦钊), Xiu-Bo Chen(陈秀波), Yu-Ping Lai(赖裕平), and Jian Li(李剑). Chin. Phys. B, 2022, 31(5): 050308.
[3] Quantum walk search algorithm for multi-objective searching with iteration auto-controlling on hypercube
Yao-Yao Jiang(姜瑶瑶), Peng-Cheng Chu(初鹏程), Wen-Bin Zhang(张文彬), and Hong-Yang Ma(马鸿洋). Chin. Phys. B, 2022, 31(4): 040307.
[4] Disorder in parity-time symmetric quantum walks
Peng Xue(薛鹏). Chin. Phys. B, 2022, 31(1): 010311.
[5] Quantum walk under coherence non-generating channels
Zishi Chen(陈子石) and Xueyuan Hu(胡雪元). Chin. Phys. B, 2021, 30(3): 030305.
[6] State transfer on two-fold Cayley trees via quantum walks
Xi-Ling Xue(薛希玲) and Yue Ruan(阮越). Chin. Phys. B, 2021, 30(2): 020304.
[7] Quantum dynamics on a lossy non-Hermitian lattice
Li Wang(王利), Qing Liu(刘青), and Yunbo Zhang(张云波). Chin. Phys. B, 2021, 30(2): 020506.
[8] High winding number of topological phase in non-unitary periodic quantum walk
Yali Jia(贾雅利) and Zhi-Jian Li(李志坚). Chin. Phys. B, 2021, 30(10): 100301.
[9] Probe of topological invariants using quantum walks of a trapped ion in coherent state space
Ya Meng(蒙雅), Feng Mei(梅锋), Gang Chen(陈刚), Suo-Tang Jia(贾锁堂). Chin. Phys. B, 2020, 29(7): 070501.
[10] A two-dimensional quantum walk driven by a single two-side coin
Quan Lin(林泉), Hao Qin(秦豪) Kun-Kun Wang(王坤坤), Lei Xiao(肖磊), and Peng Xue(薛鹏). Chin. Phys. B, 2020, 29(11): 110303.
[11] Quantum intelligence on protein folding pathways
Wen-Wen Mao(毛雯雯), Li-Hua Lv(吕丽花), Yong-Yun Ji(季永运), You-Quan Li(李有泉). Chin. Phys. B, 2020, 29(1): 018702.
[12] The entanglement of deterministic aperiodic quantum walks
Ting-Ting Liu(刘婷婷), Ya-Yun Hu(胡亚运), Jing Zhao(赵静), Ming Zhong(钟鸣), Pei-Qing Tong(童培庆). Chin. Phys. B, 2018, 27(12): 120305.
[13] Search algorithm on strongly regular graphsbased on scattering quantum walks
Xi-Ling Xue(薛希玲), Zhi-Hao Liu(刘志昊), Han-Wu Chen(陈汉武). Chin. Phys. B, 2017, 26(1): 010301.
[14] A quantum walk in phase space with resonator-assisted double quantum dots
Zhi-Hao Bian(边志浩), Hao Qin(秦豪), Xiang Zhan(詹翔), Jian Li(李剑), Peng Xue(薛鹏). Chin. Phys. B, 2016, 25(2): 020307.
[15] Localization of quantum walks on finite graphs
Yang-Yi Hu(胡杨熠), Ping-Xing Chen(陈平形). Chin. Phys. B, 2016, 25(12): 120303.
No Suggested Reading articles found!