One-dimensional lackadaisical quantum walks. (English) Zbl 1390.60171
Authors’ abstract: In this paper, we study the properties of lackadaisical quantum walks (LQWs) on a line. This model was first proposed in [T. G. Wong, J. Phys. A, Math. Theor. 48, No. 43, Article ID 435304, 17 p. (2015; Zbl 1326.81053)] as a quantum analogue of lazy random walks where each vertex is attached to \(\tau\) self-loops. We derive an analytic expression for the localization probability of the walker at the origin after infinite steps, and obtain the peak velocities of the walker. We also calculate the wave function of the walker starting from the origin and obtain a long-time approximation for the entire probability density function. As an application of the density function, we prove that lackadaisical quantum walks spread ballistically for arbitrary \(\tau\), and give an analytic solution for the variance of the walker’s probability distribution.
Reviewer: Anton Tarasenko (Novosibirsk)
MSC:
| 60G50 | Sums of independent random variables; random walks |
| 60B12 | Limit theorems for vector-valued random variables (infinite-dimensional case) |
Citations:
Zbl 1326.81053References:
| [1] | Wong T G 2015 Grover search with lackadaisical quantum walks J. Phys. A: Math. Theor.48 435304 · Zbl 1326.81053 · doi:10.1088/1751-8113/48/43/435304 |
| [2] | Aharonov Y, Davidovich L and Zagury N 1993 Quantum random walks Phys. Rev. A 48 1687 · doi:10.1103/PhysRevA.48.1687 |
| [3] | Meyer D A 1996 From quantum cellular automata to quantum lattice gases J. Stat. Phys.85 551-74 · Zbl 0952.37501 · doi:10.1007/BF02199356 |
| [4] | Farhi E and Gutmann S 1998 Quantum computation and decision trees Phys. Rev. A 58 915 · doi:10.1103/PhysRevA.58.915 |
| [5] | Spitzer F 2013 Principles of Random Walk vol 34 (Berlin: Springer) |
| [6] | Ambainis A, Bach E, Nayak A, Vishwanath A and Watrous J 2001 One-dimensional quantum walks Proc. of the 33rd Annual ACM Symp. on Theory of Computing (New York: ACM) pp 37-49 · Zbl 1323.81021 |
| [7] | Aharonov D, Ambainis A, Kempe J and Vazirani U 2001 Quantum walks on graphs Proc. of the 33rd Annual ACM Symp. on Theory of Computing (New York: ACM) pp 50-9 · Zbl 1323.81020 |
| [8] | Moore C and Russell A 2002 Quantum walks on the hypercube Randomization and Approximation Techniques in Computer Science (Berlin: Springer) pp 164-78 · Zbl 1028.68570 · doi:10.1007/3-540-45726-7_14 |
| [9] | Childs A M and Goldstone J 2004 Spatial search by quantum walk Phys. Rev. A 70 022314 · Zbl 1227.81156 · doi:10.1103/PhysRevA.70.022314 |
| [10] | Krovi H and Brun T A 2006 Hitting time for quantum walks on the hypercube Phys. Rev. A 73 032341 · doi:10.1103/PhysRevA.73.032341 |
| [11] | Aaronson S and Shi Y 2004 Quantum lower bounds for the collision and the element distinctness problems J. ACM51 595-605 · Zbl 1169.68406 · doi:10.1145/1008731.1008735 |
| [12] | Ambainis A 2007 Quantum walk algorithm for element distinctness SIAM J. Comput.37 210-39 · Zbl 1134.81010 · doi:10.1137/S0097539705447311 |
| [13] | Szegedy M 2004 Quantum speed-up of Markov chain based algorithms Proc. 45th Annual IEEE Symp. on Foundations of Computer Science (Piscataway, NJ: IEEE) pp 32-41 · doi:10.1109/FOCS.2004.53 |
| [14] | Childs A M, Cleve R, Deotto E, Farhi E, Gutmann S and Spielman D A 2003 Exponential algorithmic speedup by a quantum walk Proc. of the 35th Annual ACM Symp. on Theory of Computing (New York: ACM) pp 59-68 · Zbl 1192.81059 |
| [15] | Childs A M 2009 Universal computation by quantum walk Phys. Rev. Lett.102 180501 · doi:10.1103/PhysRevLett.102.180501 |
| [16] | Lovett N B, Cooper S, Everitt M, Trevers M and Kendon V 2010 Universal quantum computation using the discrete-time quantum walk Phys. Rev. A 81 042330 · doi:10.1103/PhysRevA.81.042330 |
| [17] | Kempe J 2003 Quantum random walks: an introductory overview Contemp. Phys.44 307-27 · doi:10.1080/00107151031000110776 |
| [18] | Venegas-Andraca S E 2012 Quantum walks: a comprehensive review Quantum Inf. Process.11 1015-106 · Zbl 1283.81040 · doi:10.1007/s11128-012-0432-5 |
| [19] | Inui N, Konno N and Segawa E 2005 One-dimensional three-state quantum walk Phys. Rev. E 72 056112 · doi:10.1103/PhysRevE.72.056112 |
| [20] | Falkner S and Boettcher S 2014 Weak limit of the three-state quantum walk on the line Phys. Rev. A 90 012307 · doi:10.1103/PhysRevA.90.012307 |
| [21] | Štefaňák M, Bezděková I and Jex I 2014 Limit distributions of three-state quantum walks: the role of coin eigenstates Phys. Rev. A 90 012342 · doi:10.1103/PhysRevA.90.012342 |
| [22] | Wang K, Wu N, Kuklinski P, Xu P, Hu H and Song F 2016 Grover walks on a line with absorbing boundaries Quantum Inf. Process.15 3573-97 · Zbl 1348.81187 · doi:10.1007/s11128-016-1353-5 |
| [23] | Inui N, Konishi Y and Konno N 2004 Localization of two-dimensional quantum walks Phys. Rev. A 69 052323 · doi:10.1103/PhysRevA.69.052323 |
| [24] | Štefaňák M, Bezděková I and Jex I 2012 Continuous deformations of the Grover walk preserving localization Eur. Phys. J. D 66 142 · doi:10.1140/epjd/e2012-30146-9 |
| [25] | Štefaňák M, Bezděková I, Jex I and Barnett S M 2014 Stability of point spectrum for three-state quantum walks on a line Quantum Inf. Comput.14 1213-26 |
| [26] | Wong T G 2017 Coined quantum walks on weighted graphs (arXiv:1703.10134) |
| [27] | Machida T 2015 Limit theorems of a 3-state quantum walk and its application for discrete uniform measures Quantum Inf. Comput.15 406-18 |
| [28] | Lyu C, Yu L and Wu S 2015 Localization in quantum walks on a honeycomb network Phys. Rev. A 92 052305 · doi:10.1103/PhysRevA.92.052305 |
| [29] | Grimmett G, Janson S and Scudo P F 2004 Weak limits for quantum random walks Phys. Rev. E 69 026119 · doi:10.1103/PhysRevE.69.026119 |
| [30] | Chandrashekar C M, Srikanth R and Laflamme R 2008 Optimizing the discrete time quantum walk using a SU(2) coin Phys. Rev. A 77 032326 · doi:10.1103/PhysRevA.77.032326 |
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