Please wait a minute...
Chin. Phys. B, 2020, Vol. 29(11): 110303    DOI: 10.1088/1674-1056/abaee8
Special Issue: SPECIAL TOPIC — Quantum computation and quantum simulation
SPECIAL TOPIC—Quantum computation and quantum simulation Prev   Next  

A two-dimensional quantum walk driven by a single two-side coin

Quan Lin(林泉)1, †, Hao Qin(秦豪)2, † Kun-Kun Wang(王坤坤)1, Lei Xiao(肖磊)1, and Peng Xue(薛鹏)1,, ‡
1 Beijing Computational Science Research Center, Beijing 100084, China
2 Department of Physics, Southeast University, Nanjing 211189, China
Abstract  

We study a two-dimensional quantum walk with only one walker alternatively walking along the horizontal and vertical directions driven by a single two-side coin. We find the analytical expressions of the first two moments of the walker’s position distribution in the long-time limit, which indicates that the variance of the position distribution grows quadratically with walking steps, showing a ballistic spreading typically for quantum walks. Besides, we analyze the correlation by calculating the quantum mutual information and the measurement-induced disturbance respectively as the outcome of the walk in one dimension is correlated to the other with the coin as a bridge. It is shown that the quantum correlation between walker spaces increases gradually with the walking steps.

Keywords:  quantum walks      quantum mutual information      measurement-induced disturbance  
Received:  30 June 2020      Revised:  04 August 2020      Accepted manuscript online:  13 August 2020
Fund: the National Natural Science Foundation of China (Grant Nos. 11674056 and U1930402) and the startup fund from Beijing Computational Science Research Center. KKW acknowledges support from the Project funded by China Postdoctoral Science Foundation (Grant No. 2019M660016).
Corresponding Authors:  These authors contributed equally. Corresponding author. E-mail: gnep.eux@gmail.com   

Cite this article: 

Quan Lin(林泉), Hao Qin(秦豪) Kun-Kun Wang(王坤坤), Lei Xiao(肖磊), and Peng Xue(薛鹏) A two-dimensional quantum walk driven by a single two-side coin 2020 Chin. Phys. B 29 110303

Fig. 1.  

Schematic for the two-dimensional QW. At the beginning of the walk, the single walker, starts from the position (x0 = 0,y0 = 0) with a two-side quantum coin in his hand. Then the walker flips the coin and takes a step along the x axis to the left or right if the coin turns out to be head or tail. Hence, the walker is possible to appear at positions (−1,0) and (1,0), i.e., the orange points. After this, the walker flips the coin again and the outcome of the coin flipping decides the walker’s movement along the y axis, up or down. Therefore, we may find the walker at positions (−1,−1), (−1,1), (1,−1), and (1,1), i.e., the yellow points in the figure with certain probabilities. We consider a step along the x axis and a step along the y axis respectively of the walker as a single course of our walk. During the second course, the walker possibly gets to the positions of the green points after the first coin flipping and finally arrives at the positions of the pink points and also the initial red point, which are 9 possible positions in total, after the second coin flipping. We omit the intermediate positions the walker may reach from the third course of the walk in the figure. It can be seen that after the n-th course the walk shows a spreading of position distribution including (n + 1)2 possible points on the two-dimensional lattice.

Fig. 2.  

(a) The position distribution of the two-dimensional CRW after the 10th step. (b) The variance of the position distributions of the two-dimensional CRW for the fist 10 steps.

Fig. 3.  

(a) The position distribution of the two-dimensional QW after the 10th step. (b) The variance of the position distributions of the two-dimensional QW for the fist 10 steps.

Fig. 4.  

(a) The quantum mutual information of two-dimensional QW (squares) and of the maximal entangled state (dots). (b) The ln–ln plot of the quantum mutual information of two-dimensional QW (squares) and of the maximal entangled state (dots).

Fig. 5.  

(a) The measurement-induced disturbance of two-dimensional QW (squares) and of the maximal entangled state (dots). (b) The ln–ln plot of the measurement-induced disturbance of two-dimensional QW (squares) and of the maximal entangled state (dots).

[1]
Dür W, Raussendorf R, Kendon V M, Briegel H J 2002 Phys. Rev. A 66 052319 DOI: 10.1103/PhysRevA.66.052319
[2]
Ambainis A 2003 Int. J. Quantum Inf. 1 507 DOI: 10.1142/S0219749903000383
[3]
Childs A M, Cleve R, Deotto E, Farhi E, Gutmann S, Spielman D A 2003 Proc. 35th ACM Symposium on Theory of Computing 59 68 DOI: 10.1145/780542.780552
[4]
Shenvi N, Kempe J, Whaley K B 2003 Phys. Rev. A 67 052307 DOI: 10.1103/PhysRevA.67.052307
[5]
Kempe J 2003 Contempo. Phys. 44 307 DOI: 10.1080/00107151031000110776
[6]
Izaac J A, Zhan X, Bian Z H, Wang K K, Li J, Wang J B, Xue P 2017 Phys. Rev. A 95 032318 DOI: 10.1103/PhysRevA.95.032318
[7]
Childs A M 2009 Phys. Rev. Lett. 102 180501 DOI: 10.1103/PhysRevLett.102.180501
[8]
Childs A M, Gosset D, Webb Z 2013 Science 339 791 DOI: 10.1126/science.1229957
[9]
Xue P, Zhang R, Qin H, Zhan X, Bian Z H, Li J, Sanders B C 2015 Phys. Rev. Lett. 114 140502 DOI: 10.1103/PhysRevLett.114.140502
[10]
Kitagawa T, Rudner M S, Berg E, Demler E 2010 Phys. Rev. A 82 033429 DOI: 10.1103/PhysRevA.82.033429
[11]
Oliveira A C, Portugal R, Donangelo R 2006 Phys. Rev. A 74 012312 DOI: 10.1103/PhysRevA.74.012312
[12]
Xue P, Sanders B C 2013 Phys. Rev. A 87 022334 DOI: 10.1103/PhysRevA.87.022334
[13]
Zhang R, Xue P, Twamley J 2014 Phys. Rev. A 89 042317 DOI: 10.1103/PhysRevA.89.042317
[14]
Zhan X, Qin H, Bian Z H, Li J, Xue P 2014 Phys. Rev. A 90 012331 DOI: 10.1103/PhysRevA.90.012331
[15]
Xiao L, Zhan X, Bian Z H, Wang K K, Zhang X, Wang X P, Li J, Mochizuki K, Kim D, Kawakami N, Yi W, Obuse H, Sanders B C, Xue P 2017 Nat. Phys. 13 1117 DOI: 10.1038/nphys4204
[16]
Xiao L, Deng T S, Wang K K, Zhu G Y, Wang Z, Yi W, Xue P 2020 Nat. Phys. 16 761 DOI: 10.1038/s41567-020-0836-6
[17]
Kurzyński P, Wójcik A 2013 Phys. Rev. Lett. 110 200404 DOI: 10.1103/PhysRevLett.110.200404
[18]
Bian Z H, Li J, Qin H, Zhan X, Zhang R, Sanders B C, Xue P 2015 Phys. Rev. Lett. 114 203602 DOI: 10.1103/PhysRevLett.114.203602
[19]
Brun T A, Carteret H A, Ambainis A 2003 Phys. Rev. Lett. 91 130602 DOI: 10.1103/PhysRevLett.91.130602
[20]
Cardano F, Massa F, Qassim H, Karimi E, Slussarenko S, Paparo D, de Lisio C, Sciarrino F, Santamato E, Boyd R W, Marrucci L 2015 Sci. Adv. 1 e1500087 DOI: 10.1126/sciadv.1500087
[21]
Defienne H, Barbieri M, Walmsley I A, Smith B J, Gigan S 2016 Sci. Adv. 1 e1501054 DOI: 10.1126/sciadv.1501054
[22]
Wang K K, Qiu X, Xiao L, Zhan X, Bian Z H, Yi W, Xue P 2019 Phys. Rev. Lett. 122 020501 DOI: 10.1103/PhysRevLett.122.020501
[23]
Du J, Li H, Xu X, Shi M, Wu J, Zhou X, Han R 2003 Phys. Rev. A 67 042316 DOI: 10.1103/PhysRevA.67.042316
[24]
Ryan C A, Laforest M, Boileau J C, Laflamme R 2005 Phys. Rev. A 72 062317 DOI: 10.1103/PhysRevA.72.062317
[25]
Broome M A, Fedrizzi A, Lanyon B P, Kassal I, Aspuru-Guzik A, White A G 2010 Phys. Rev. Lett. 104 153602 DOI: 10.1103/PhysRevLett.104.153602
[26]
Schreiber A, Cassemiro K N, Potocek V, Gabris A, Mosley P J, Andersson E, Jex I, Silberhorn C 2010 Phys. Rev. Lett. 104 050502 DOI: 10.1103/PhysRevLett.104.050502
[27]
Kitagawa T, Broome M A, Fedrizzi A, Rudner M S, Berg E, Kassal I, Aspuru-Guzik A, Demler E, White A G 2012 Nat. Commun. 3 882 DOI: 10.1038/ncomms1872
[28]
Qin H, Xue P 2016 Chin. Phys. B 25 010501 DOI: 10.1088/1674-1056/25/1/010501
[29]
Bian Z H, Qin H, Zhan Z, Li J, Xue P 2016 Chin. Phys. B 25 020307 DOI: 10.1088/1674-1056/25/2/020307
[30]
Luo H, Zhan Z, Zhang R, Xue P 2016 Chin. Phys. B 25 110304 DOI: 10.1088/1674-1056/25/11/110304
[31]
Zhang R, Xu Y, Xue P 2015 Chin. Phys. B 24 010303 DOI: 10.1088/1674-1056/24/1/010303
[32]
Xue P, Qin H, Zhan Z, Bian Z H, Li J 2014 Chin. Phys. B 23 110307 DOI: 10.1088/1674-1056/23/11/110307
[33]
Qin H, Xue P 2014 Chin. Phys. B 23 010301 DOI: 10.1088/1674-1056/23/1/010301
[34]
Mackay T D, Bartlett S D, Stephenson L T, Sanders B C 2002 J. Phys. A 35 2745 DOI: 10.1088/0305-4470/35/12/304
[35]
Franco C Di, Gettrick M Mc, Busch Th 2011 Phys. Rev. Lett. 106 080502 DOI: 10.1103/PhysRevLett.106.080502
[36]
Franco C Di, Gettrick M Mc, Machida T, Busch Th 2011 Phys. Rev. A 84 042337 DOI: 10.1103/PhysRevA.84.042337
[37]
Brun T A, Carteret H A, Ambainis A 2003 Phys. Rev. A 67 052317 DOI: 10.1103/PhysRevA.67.052317
[38]
Stefanak M, Barnett S M, Kollar B, Kiss T, Jex I 2011 New J. Phys. 13 033029 DOI: 10.1088/1367-2630/13/3/033029
[39]
Rohde P P, Schreiber A, Stefanak M, Jex I, Silberhorn C 2001 New J. Phys. 13 013001 DOI: 10.1088/1367-2630/13/1/013001
[40]
Ahlbrecht A, Vogts H, Werner A H, Werner R F 2011 J. Math. Phys. 52 042201 DOI: 10.1063/1.3575568
[41]
Berry S D, Wang J B 2011 Phys. Rev. A 83 042317 DOI: 10.1103/PhysRevA.83.042317
[42]
Zahringer F, Kirchmair G, Gerritsma R, Solano E, Blatt R, Roos C F 2010 Phys. Rev. Lett. 104 100503 DOI: 10.1103/PhysRevLett.104.100503
[43]
Peruzzo A, Lobino M, Matthews J C F, Matsuda N, Politi A, Poulios K, Zhou X, Lahini Y, Ismail N, Worhoff K, Bromberg Y, Silberberg Y, Thompson M G, OBrien J L 2010 Science 329 1500 DOI: 10.1126/science.1193515
[44]
Schreiber A, Gabris A, Rohde P P, Laiho K, Stefanak M, Potocek V, Hamilton C, Jex I, Silberhorn C 2012 Science 336 55 DOI: 10.1126/science.1218448
[45]
Crespi A, Osellame R, Ramponi R, Giovannetti V, Fazio R, Sansoni L, Nicola F D, Sciarrino F, Mataloni P 2013 Nat. Photon. 7 322 DOI: 10.1038/nphoton.2013.26
[46]
Mohseni M, Rebentrost P, Lloyd S, Aspuru-Guzik A 2008 J. Chem. Phys. 129 174106 DOI: 10.1063/1.3002335
[47]
Sension R J 2007 Nature 446 740 DOI: 10.1038/446740a
[48]
Wang B, Chen T, Zhang X 2018 Phys. Rev. Lett. 121 100501 DOI: 10.1103/PhysRevLett.121.100501
[49]
DErrico A, Cardano F, Maffei M, Dauphin A, Barboza R, Esposito C, Piccirillo B, Lewenstein M, Massignan P, Marrucci L 2020 Optica 7 108 DOI: 10.1364/OPTICA.365028
[50]
Nayak A, Vishwanath A 2000 e-print quant-ph/0010117
[51]
Groisman B, Popescu S, Winter A 2005 Phys. Rev. A 72 032317 DOI: 10.1103/PhysRevA.72.032317
[52]
Ollivier H, Zurek W H 2001 Phys. Rev. Lett. 88 017901 DOI: 10.1103/PhysRevLett.88.017901
[53]
Luo S 2008 Phys. Rev. A 77 022301 DOI: 10.1103/PhysRevA.77.022301
[54]
Modi K, Paterek T, Son W, Vedral V, Williamson M 2010 Phys. Rev. Lett. 104 080501 DOI: 10.1103/PhysRevLett.104.080501
[55]
Vedral V, Plenio M B, Rippin M A, Knight P L 1997 Phys. Rev. Lett. 78 2275 DOI: 10.1103/PhysRevLett.78.2275
[1] 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.
[2] 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.
[3] 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.
[4] Quantum walks with coins undergoing different quantum noisy channels
Hao Qin(秦豪) and Peng Xue(薛鹏). Chin. Phys. B, 2016, 25(1): 010501.
[5] Quantum correlations in a two-qubit anisotropic Heisenberg XYZ chain with uniform magnetic field
Li Lei, Yang Guo-Hui. Chin. Phys. B, 2014, 23(7): 070306.
[6] Measurement-induced disturbance in Heisenberg XY spin model with Dzialoshinskii-Moriya interaction under intrinsic decoherence
Shen Cheng-Gao, Zhang Guo-Feng, Fan Kai-Ming, Zhu Han-Jie. Chin. Phys. B, 2014, 23(5): 050310.
[7] Correlation dynamics of a qubit–qutrit system in a spin-chain environment with Dzyaloshinsky–Moriya interaction
Yang Yang, Wang An-Min. Chin. Phys. B, 2014, 23(2): 020307.
[8] Quantum correlation switches for dipole arrays
Li Yan-Jie, Liu Jin-Ming, Zhang Yan. Chin. Phys. B, 2014, 23(11): 110306.
[9] Measurement-induced disturbance between two atoms in Tavis–Cummings model with dipole–dipole interaction
Zhang Guo-Feng, Wang Xiao, Lü Guang-Hong. Chin. Phys. B, 2014, 23(10): 104204.
[10] Implementation of a one-dimensional quantum walk in both position and phase spaces
Qin Hao, Xue Peng. Chin. Phys. B, 2014, 23(1): 010301.
[11] Non-Markovian decoherent quantum walks
Xue Peng, Zhang Yong-Sheng. Chin. Phys. B, 2013, 22(7): 070302.
[12] Correlation dynamics of two-parameter qubit-qutrit states under decoherence
Yuan Hao, Wei Lian-Fu. Chin. Phys. B, 2013, 22(5): 050303.
[13] Measurement-induced disturbance and nonequilibrium thermal entanglement in a qutrit–qubit mixed spin XXZ model
Chen Li, Shao Xiao-Qiang, Zhang Shou. Chin. Phys. B, 2011, 20(10): 100311.
No Suggested Reading articles found!