1887
banner image
No data available.
Please log in to see this content.
You have no subscription access to this content.
No metrics data to plot.
The attempt to load metrics for this article has failed.
The attempt to plot a graph for these metrics has failed.
The full text of this article is not currently available.
f
Asymptotic evolution of quantum walks with random coin
Rent:
Rent this article for
Access full text Article
/content/aip/journal/jmp/52/4/10.1063/1.3575568
1.
1.A. Ambainis, E. Bach, A. Nayak, A. Vishwanath, and J. Watrous, Proc. 33rd ACM Stoc (2001).
2.
2.D. A. Meyer, “From quantum cellular automata to quantum lattice gases,” J. Stat. Phys. 85, 551 (1996).
http://dx.doi.org/10.1007/BF02199356
3.
3.M. Karski, L. Förster, J.-M. Choi, A. Steffen, W. Alt, D. Meschede, and A. Widera, “Quantum walk in position space with single optically trapped atoms,” Science 325, 174 (2009).
http://dx.doi.org/10.1126/science.1174436
4.
4.M. McGettrick, “One dimensional quantum walks with memory,” Quantum Inf. Comput. 10, 0509 (2010).
5.
5.H. Vogts, “Discrete time quantum lattice systems,” Ph.D. dissertation, Technische Universität Braunschweig (2009).
6.
6.H. A. Carteret, M. E. H. Ismail, and B. Richmond, “Three routes to the exact asymptotics for the one-dimensional quantum walk,” J. Phys. A 36, 8775 (2003).
http://dx.doi.org/10.1088/0305-4470/36/33/305
7.
7.G. Grimmett, S. Janson, and P. F. Scudo, “Weak limits for quantum random walks,” Phys. Rev. E 69, 026119 (2004).
http://dx.doi.org/10.1103/PhysRevE.69.026119
8.
8.N. Konno, “Quantum random walks in one dimension,” Quantum Inf. Proc. 1, 345 (2002).
http://dx.doi.org/10.1023/A:1023413713008
9.
9.N. Konno, “A new type of limit theorems for the one-dimensional quantum random walk,” J. Math. Soc. Jpn. 57, 1179 (2005).
http://dx.doi.org/10.2969/jmsj/1150287309
10.
10.A. Bressler and R. Pemantle, “Quantum random walks in one dimension via generating functions,” in Proceedings of the 2007 Conference on Analysis of Algorithms, Juan des Pins, France (DMTCS Proceedings, 2007).
11.
11.T. Mackay, S. D. Bartlett, L. T. Stephenson, and B. C. Sanders, “Quantum walks in higher dimensions,” J. Phys. A 35, 2745 (2002).
http://dx.doi.org/10.1088/0305-4470/35/12/304
12.
12.B. Kollár, M. Štefaňák, T. Kiss, and I. Jex, “Recurrences in three-state quantum walks on a plane,” Phys. Rev. A 82, 012303 (2010).
http://dx.doi.org/10.1103/PhysRevA.82.012303
13.
13.Y. Baryshnikov, W. Brady, A. Bressler, and R. Pemantle, “Two-dimensional quantum random walk,” J. Stat. Phys. 142, 78 (2010).
http://dx.doi.org/10.1007/s10955-010-0098-2
14.
14.A. Bressler, T. Greenwood, R. Pemantle, and M. Petkovšek, “Quantum random walk on the integer lattice: examples and phenomena,” Algorithmic Probability and Combinatorics, Contemporary Mathematics, Vol. 520, (AMS, 2010), pp. 4160.
15.
15.V. Kendon, “Decoherence in quantum walks - a review,” Math. Struct. Comput. Sci. 17 (6), 1169 (2006).
16.
16.A. Romanelli, “Measurements in the Lévy quantum walk,” Phys. Rev. A 76, 054306 (2007).
http://dx.doi.org/10.1103/PhysRevA.76.054306
17.
17.A. Romanelli, R. Siri, G. Abal, A. Auyuanet, and R. Donangelo, “Decoherence in the quantum walk on the line,” Phys. A: Stat. Mech. Appl. 347, 137 (2005).
http://dx.doi.org/10.1016/j.physa.2004.08.070
18.
18.R. Srikanth, S. Banerjee, and C. M. Chandrashekar, “Quantumness in decoherent quantum walk using measurement-induced disturbance,” Phys. Rev. A 81, 062123 (2010).
http://dx.doi.org/10.1103/PhysRevA.81.062123
19.
19.K. Zhang, “Limiting distribution of decoherent quantum random walks,” Phys. Rev. A 77, 062302 (2008).
http://dx.doi.org/10.1103/PhysRevA.77.062302
20.
20.K. C. N. Konno, E. Segawa, and Y. Shikano, “Randomness and arrow of time in quantum walks,” Phys. Rev. A 81, 062129 (2010).
http://dx.doi.org/10.1103/PhysRevA.81.062129
21.
21.G. Abal, R. Donangelo, F. Severo, and R. Siri, “Decoherent quantum walks driven by a generic coin operation,” Phys. A: Stat. Mech. Appl. 387, 335 (2007).
http://dx.doi.org/10.1016/j.physa.2007.08.058
22.
22.D. Shapira, O. Biham, A. Bracken, and M. Hackett, “One dimensional quantum walk with unitary noise,” Phys. Rev. A 68, 062315 (2003).
http://dx.doi.org/10.1103/PhysRevA.68.062315
23.
23.C. Chandrashekar, R. Srikanth, and S. Banerjee, “Symmetries and noise in quantum walk,” Phys. Rev. A 76, 022316 (2007).
http://dx.doi.org/10.1103/PhysRevA.76.022316
24.
24.T. A. Brun, H. A. Carteret, and A. Ambainis, “Quantum random walks with decoherent coins,” Phys. Rev. A 67, 032304 (2003).
http://dx.doi.org/10.1103/PhysRevA.67.032304
25.
25.T. A. Brun, H. A. Carteret, and A. Ambainis, “Quantum walks driven by many coins,” Phys. Rev. A 67, 052317 (2002).
http://dx.doi.org/10.1103/PhysRevA.67.052317
26.
26.T. A. Brun, H. A. Carteret, and A. Ambainis, “The quantum to classical transition for random walks,” Phys. Rev. Lett. 91, 130602 (2003).
http://dx.doi.org/10.1103/PhysRevLett.91.130602
27.
27.E. Segawa and N. Konno, “Limit theorems for quantum walks driven by many coins,” Int. J. Quantum. Inf. 6, 1231 (2008).
http://dx.doi.org/10.1142/S0219749908004456
28.
28.J. Košík, V. Bužek, and M. Hillery, “Quantum walks with random phase shifts,” Phys. Rev. A 74, 022310 (2006).
http://dx.doi.org/10.1103/PhysRevA.74.022310
29.
29.M. Annabestani, S. J. Akhtarshenas, and M. R. Abolhassani, “Decoherence in one-dimensional quantum walk,” Phys. Rev. A 81, 032321 (2010).
http://dx.doi.org/10.1103/PhysRevA.81.032321
30.
30.M. Annabestani, S. J. Akhtarshenas, and M. R. Abolhassani, “Tunneling effects in a one-dimensional quantum walk,” e-print arXiv:quant-ph/1004.4352 (2010).
31.
31.G. Leung, P. Knott, J. Bailey, and V. Kendon, “Coined quantum walks on percolation graphs,” New J. Phys. 12, 123018 (2010).
http://dx.doi.org/10.1088/1367-2630/12/12/123018
32.
32.N. Konno, “A path integral approach for disordered quantum walks in one dimension,” Fluct. Noise Lett. 5, 529 (2005).
http://dx.doi.org/10.1142/S0219477505002987
33.
33.O. Bratteli and P. E. T. Jorgensen, “Wavelet filters and infinite-dimensional unitary groups,” in Proceedings of the International Conference on Wavelet Analysis and Application, AMS/IP Studies in Advanced Mathematics, Vol. 25, (AMS/International Press, New York, 2002), pp. 3565.
34.
34.P. P. Vaidyanathan and Z. Doǧanata, “The role of lossless systems in modern digital signal processing: A tutorial,” IEEE Trans. Education 32, 181 (1989).
http://dx.doi.org/10.1109/13.34150
35.
35.T. Q. Nguyen, X. Gao, and G. Strang, “On factorization of m-channel paraunitary filterbanks,” IEEE Trans. Signal Process. 49(7), 1433 (2001).
http://dx.doi.org/10.1109/78.928696
36.
36.D. Gross, V. Nesme, H. Vogts, and R. Werner, “Index theory of one dimensional quantum walks and cellular automata,” Commun. Math. Phys. (to appear) e-print arXiv:quant-ph/0910.3675 (2009).
37.
37.A. Ahlbrecht, V. B. Scholz, and A. H. Werner, “Disordered quantum walks in one lattice dimensions,” e-print arXiv:quant-ph/1101.2298v2 (2011).
38.
38.W. F. Stinespring, “Positive functions on C*-algebras,” Proc. Am. Math. Soc. 6, 211 (1955).
39.
39.V. Paulsen, Completely Bounded Maps on Operator Algebras (Cambridge University Press, Cambridge, England, 2002).
40.
40.K. Kraus, Lecture Notes in Physics (Springer, New York, 1983).
41.
41.T. Kato, Perturbation Theory for Linear Operators (Springer, New York, 1995).
42.
42.A. Ambainis, “Quantum walks and their algorithmic application,” Int. J. Quantum Inf. 1, 507 (2003).
http://dx.doi.org/10.1142/S0219749903000383
43.
43.J. Kempe, “Quantum random walks: An introductory overview,” Contemp. Phys. 44, 307 (2003).
http://dx.doi.org/10.1080/00107151031000110776
http://aip.metastore.ingenta.com/content/aip/journal/jmp/52/4/10.1063/1.3575568
Loading
/content/aip/journal/jmp/52/4/10.1063/1.3575568
Loading

Data & Media loading...

Loading

Article metrics loading...

/content/aip/journal/jmp/52/4/10.1063/1.3575568
2011-04-19
2015-04-25

Abstract

We study the asymptotic position distribution of general quantum walks on a lattice, including walks with a random coin, which is chosen from step to step by a general Markov chain. In the unitary (i.e., nonrandom) case, we allow any unitary operator which commutes with translations and couples only sites at a finite distance from each other. For example, a single step of the walk could be composed of any finite succession of different shift and coin operations in the usual sense, with any lattice dimension and coin dimension. We find ballistic scaling and establish a direct method for computing the asymptotic distribution of position divided by time, namely as the distribution of the discrete time analog of the group velocity. In the random case, we let a Markov chain (control process) pick in each step one of finitely many unitary walks, in the sense described above. In ballistic order, we find a nonrandom drift which depends only on the mean of the control process and not on the initial state. In diffusive scaling, the limiting distribution is asymptotically Gaussian, with a covariance matrix (diffusion matrix) depending on momentum. The diffusion matrix depends not only on the mean but also on the transition rates of the control process. In the nonrandom limit, i.e., when the coins chosen are all very close or the transition rates of the control process are small, leading to long intervals of ballistic evolution, the diffusion matrix diverges. Our method is based on spatial Fourier transforms, and the first and second order perturbation theory of the eigenvalue 1 of the transition operator for each value of the momentum.

Loading

Full text loading...

/deliver/fulltext/aip/journal/jmp/52/4/1.3575568.html;jsessionid=f9tk0cfkgkc3r.x-aip-live-06?itemId=/content/aip/journal/jmp/52/4/10.1063/1.3575568&mimeType=html&fmt=ahah&containerItemId=content/aip/journal/jmp
true
true
This is a required field
Please enter a valid email address

Oops! This section, does not exist...

Use the links on this page to find existing content.

752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Asymptotic evolution of quantum walks with random coin
http://aip.metastore.ingenta.com/content/aip/journal/jmp/52/4/10.1063/1.3575568
10.1063/1.3575568
SEARCH_EXPAND_ITEM