Localization of quantum walks on finite graphs

doi:10.1088/1674-1056/25/12/120303

Abstract

We analyze the localization of quantum walks on a one-dimensional finite graph using vector-distance. We first vectorize the probability distribution of a quantum walker in each node. Then we compute out the probability distribution vectors of quantum walks in infinite and finite graphs in the presence of static disorder respectively, and get the distance between these two vectors. We find that when the steps taken are small and the boundary condition is tight, the localization between the infinite and finite cases is greatly different. However, the difference is negligible when the steps taken are large or the boundary condition is loose. It means quantum walks on a one-dimensional finite graph may also suffer from localization in the presence of static disorder. Our approach and results can be generalized to analyze the localization of quantum walks in higher-dimensional cases.

Keywords: localization of quantum walks vector distance static disorder boundary conditions

Received: 12 May 2016
Revised: 17 August 2016
Accepted manuscript online:

PACS:

03.67.Ac

(Quantum algorithms, protocols, and simulations)

Fund:

Project supported by the National Natural Science Foundation of China (Grant No. 11174370).

Corresponding Authors: Ping-Xing Chen
E-mail: pxchen@nudt.edu.cn

Cite this article:

Yang-Yi Hu(胡杨熠), Ping-Xing Chen(陈平形) Localization of quantum walks on finite graphs 2016 Chin. Phys. B 25 120303

[1]	Kempe J 2003 Contemporary Physics 44 307
[2]	Rakovszky T and Asboth J K 2015 Phys. Rev. A 92 052311
[3]	Shenvi N, Kempe J and Whaley K B 2003 Phys. Rev. A 67 052307
[4]	Lovett N B, Cooper S, Everitt M, Trevers M and Kendon V 2010 Phys. Rev. A 81 042330
[5]	Travaglione B C and Milburn G J 2002 Phys. Rev. A 65 032310
[6]	Zahringer F, Kirchmair G, Gerritsma R, Solano E, Blatt R and Roos C F 2010 Phys. Rev. Lett. 104 100503
[7]	Schmitz H, Matjeschk R, Schneider C, Glueckert J, Enderlein M, Huber T and Schaetz T 2009 Phys. Rev. Lett. 103 090504
[8]	Karski M, Forster L, Choi J M, Steken A, Alt W, Meschede D and Widera A 2009 Science 325 174
[9]	Genske M, Alt W, Steken A, Werner A H, Werner R F, Meschede D and Alberti A 2013 Phys. Rev. Lett. 110 190601
[10]	Robens C, Alt W, Meschede D, Emary C and Alberti A 2015 Phys. Rev. X 5 011003
[11]	Schreiber A, Cassemiro K N, Pototcek V, Gsabris A, Mosley P J, Andersson E, Jex I and Silberhorn C 2010 Phys. Rev. Lett. 104 050502
[12]	Schreiber A, Gsabris A, Rohde P P, Laiho K, TStefatnsak M, Pototcek V, Hamilton C, Jex I and Silberhorn C 2012 Science 336 55
[13]	Peruzzo A, Lobino M, Matthews J C F, Matsuda N, Politi A, Poulios K, Zhou X Q, Lahini Y, Ismail N, Wrhok K, Bromberg Y, Silberberg Y, Thompson M G and O'Brien J L 2010 Science 329 1500
[14]	Broome M A, Fedrizzi A, Lanyon B P, Kassal I, Aspuru-Guzik A and White A G 2010 Phys. Rev. Lett. 104 153602
[15]	Sansoni L, Sciarrino F, Vallone G, Mataloni P, Crespi A, Ramponi R and Osellame R 2012 Phys. Rev. Lett. 108 010502
[16]	Zhao Y Y, Yu N K, Kurzynski P, Xiang G Y, Li C F and Guo G C 2015 Phys. Rev. A 91 042101
[17]	Ct R, Russell A, Eyler E E and Gould P L 2006 New J. Phys. 8 156
[18]	Ksalmsan O, Kiss T and Foldi P 2009 Phys. Rev. B 80 035327
[19]	Joye A and Merkli M 2010 J. Stat. Phys. 140 1025
[20]	Ahlbrecht A, Scholz V B and Werner A H 2011 J. Math. Phys. 52 102201
[21]	De Nicola F, Sansoni L, Crespi A, Ramponi R, Osellame R, Giovannetti V, Fazio R, Mataloni P and Sciarrino F 2014 Phys. Rev. A 89 032322
[22]	Schreiber A, Cassemiro K N, Pototcek V, Gsabris A, Jex I and Silberhorn C 2011 Phys. Rev. Lett. 106 180403
[23]	Romanelli A, Auyuanet A, Siri R, Abal G and Donangelo R 2005 Physica A 352 409
[24]	Oka T, Konno N, Arita R and Aoki H 2005 Phys. Rev. Lett. 94 100602
[25]	Yin Y, Katsanos D E and Evangelou S N 2008 Phys. Rev. A 77 022302
[26]	Konno N 2010 Quantum Inform. Process. 9 405
[27]	Chandrashekar C M 2009 "Discrete-Time Quantum Walk–Dynamics and Applications", Ph. D. Thesis, University of Waterloo
[28]	Shikano Y and Katsura H 2010 Phys. Rev. E 82 031122
[29]	Chandrashekar C M, Goyal S K and Banerjee S J 2012 Quantum Inform. Sci. 2 15
[30]	Chandrashekar C M 2010 Phys. Rev. A 83 022320
[31]	Farhi E and Gutmann S 1998 Phys. Rev. A 58 915
[32]	Aharonov Y, Davidovich L and Zagury N 1993 Phys. Rev. A 48 1687
[33]	Meyer D A 1996 J. Stat. Phys. 85 551
[34]	Ambainis A, Bach E, Nayak A, Vishwanath A and Watrous J 2001 Proceedings of the 33rd ACM Symposium on Theory of Computing (New York:ACM Press), p. 60
[35]	Nayak A and Vishwanath A D 2001 Technical Report, No. 2000-43, arXiv:quant-ph/0010117
[36]	Ambainis A 2003 Int. J. Quantum Inform. 1 507
[37]	Childs A M, Cleve E, Deotto E, Farhi E, Gutmann S and Spielman D A 2003 Proceedings of the 35th ACM Symposium on Theory of Computing, (New York:ACM Press), p. 59
[38]	Shenvi N, Kempe J and Whaley K B 2003 Phys. Rev. A 67 052307
[39]	Ambainis A, Kempe J, and Rivosh A 2005 Proceedings of ACM-SIAM Symp. on Discrete Algorithms (SODA), (New York:AMC Press), pp. 1099-1108, ISBN:0-89871-585-7

[1]	Riemann-Hilbert approach and N double-pole solutions for a nonlinear Schrödinger-type equation Guofei Zhang(张国飞), Jingsong He(贺劲松), and Yi Cheng(程艺). Chin. Phys. B, 2022, 31(11): 110201.
[2]	Steered coherence and entanglement in the Heisenberg XX chain under twisted boundary conditions Yu-Hang Sun(孙宇航) and Yu-Xia Xie(谢玉霞). Chin. Phys. B, 2021, 30(7): 070303.
[3]	Insights into the regulation mechanism of ring-shaped magnetoelectric energy harvesters via mechanical and magnetic conditions Yang Shi(师阳), Ni Li(李妮), and Yong Yang(杨勇). Chin. Phys. B, 2021, 30(10): 107503.
[4]	Influence of electron correlations on double-capture process in proton helium collisions Hoda Ghavaminia, Ebrahim Ghanbari-Adivi. Chin. Phys. B, 2015, 24(7): 073401.
[5]	Projectile angular-differential cross sections for single electron transfer in fast He⁺-He collisions Ebrahim Ghanbari-Adivi, Hoda Ghavaminia. Chin. Phys. B, 2015, 24(3): 033401.
[6]	Recursion-transform approach to compute the resistance of a resistor network with an arbitrary boundary Tan Zhi-Zhong (谭志中). Chin. Phys. B, 2015, 24(2): 020503.
[7]	MHD flow of nanofluids over an exponentially stretching sheet in a porous medium with convective boundary conditions T. Hayat, M. Imtiaz, A. Alsaedi, R. Mansoor. Chin. Phys. B, 2014, 23(5): 054701.
[8]	A density functional theory study on parameters fitting of ultra long armchair (n, n) single walled boron nitride nanotubes Wang Yan-Li(王艳丽), Zhang Jun-Ping(张军平), Su Ke-He(苏克和), Wang Xin(王欣), Liu Yan(刘艳), and Sun Xu(孙旭) . Chin. Phys. B, 2012, 21(6): 060301.
[9]	Real scalar field scattering with polynomial approximation around Schwarzschild--de Sitter black-hole Liu Mo-Lin(刘墨林), Liu Hong-Ya(刘宏亚), Zhang Jing-Fei(张敬飞), and Yu Fei(于飞). Chin. Phys. B, 2008, 17(5): 1633-1639.
[10]	GaN-based heterostructures: electric--static equilibrium and boundary conditions Zhang Jin-Feng(张金风) and Hao Yue(郝跃). Chin. Phys. B, 2006, 15(10): 2402-2406.
[11]	Non-equilibrium extrapolation method for velocity and pressure boundary conditions in the lattice Boltzmann method Guo Zhao-Li (郭照立), Zheng Chu-Guang (郑楚光), Shi Bao-Chang (施保昌). Chin. Phys. B, 2002, 11(4): 366-374.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

Localization of quantum walks on finite graphs

Cite this article:

Online attention