Limit theorems for quantum walks driven by many coins

International Journal of Quantum Information

Volumn 6, Issue 6, 2008, Pages 1231-1243

  Segawa, Etsuo ^a   Konno, Norio ^a  

^a YOKOHAMA NATIONAL UNIVERSITY   (Japan)

Author keywords

Limit theorem; Many coins; Quantum walks

Indexed keywords

A-PARTICLES; CLASSICAL BEHAVIOR; LIMIT THEOREM; MANY COINS; QUANTUM WALK; TIME STEP;

EID: 58149398152     PISSN: 02197499     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0219749908004456     Document Type: Article

References (32)

1. D. A. Meyer, From quantum cellular automata to quantum lattice gases, J. Statist. Phys. 85 (1996) 551-574.
2. Y. Aharonov, L. Daidovich and N. Zagury, Quantum random walks, Phys. Rev. A 48 (1993) 1689.
3. D. Ahronov, A. Ambainis, J. Kempe and U. Vazirani, Quantum walks on graphs, in Proc. 33th ACM Symposium on Theory of Computing (2001), pp. 50-59
4. N. Konno, Limit theorems and absorption problems for one-dimensional correlated random walks, quant-ph/0310191.
5. N. Shenvi, J. Kempe and B. Whaley, A quantum random walk search algorithm, Phys. Rev. A 67 (2003) 052307.
6. A. Ambainis, Quantum walk algorithm for element distinctness, in Proc. 45th IEEE Symp. Foundations of Computer Science (2004), pp. 22-31, also appears in SIAM J Comput. 31 (2007) 210-239
7. F. Magniez, A. Nayak, J. Roland and M. Santha, Search via quantum walk, in Proc. 39th ACM Symp. Theory of Computing (2007), pp. 575-584.
8. N. Konno, Limit theorems and absorption problems for quantum random walks in one dimension, Quant. Inform. Comput. 2 (2002) 578-595.
9. T. Oka, N. Konno, R. Arita and H. Aoki, Breakdown of an electric-field driven system: A mapping to a quantum walk, Phys. Rev. Lett. 94 (2005) 100602.
10. J. Kempe, Quantum random walk - An introductory overview, Contemp. Phys. 44 (2003) 307-327.
11. A. Ambainis, Quantum walks and their algorithmic applications, Int. J. Quant. Inform. 1 (2003) 507-518.
12. V. Kendon, Decoherence in quantum walks - A review, Math. Struct. Comput. Sci. 17 (2007) 1169-1220.
13. N. Konno, Quantum walks, Lecture Notes in Mathematics, Vol. 1954, eds. U. Franz and M. Schürmann (Springer, 2008), pp. 309-452.
14. N. Konno, Quantum random walks in one dimension, Quant. Inform. Process. 1 (2002) 345-354.
15. N. Konno, A new type of limit theorems for the one-dimensional quantum random walk, J. Math. Soc. Japan 57 (2005) 1179-1195.
16. G. Grimmett, S. Janson and P. Scudo, Weak limits for quantum random walks, Phys. Rev. E 69 (2004) 026119.
17. T. A. Brun, H. A. Carteret and A. Ambainis, Quantum to classical transition for random walks, Phys. Rev. Lett. 91 (2003) 130602.
18. T. A. Brun, H. A. Carteret and A. Ambainis, Quantum walks driven by many coins, Phys. Rev. A 67 (2003) 062317.
19. A. P. Flitney and D. Abbott, Quantum random walk with history dependence, J. Phys. A: Math. Gen. 37 (2004) 7581-7591.
20. S. Severini, Universal quantum computation with unlabeled qubits, J. Phys. A: Math. Gen. 39 (2006) 8507-8515.
21. M. A. Jafarizadeh and S. Salimi, Investigation of continuous-time quantum walk via spectral distribution associated with adjacency matrix, Ann. Phys. 322 (2007) 1005-1033.
22. N. Konno, Limit theorem for continuous-time quantum walk on the line, Phys. Rev. E 72 (2005) 026113.
23. N. Konno, Continuous-time quantum walk on trees in quantum probability theory, Infinite Dimensional Anal. Quant. Probab. Related Top. 9 (2006) 287-297.
24. N. Konno, Continuous-time quantum walks on ultrametric spaces, Int. J. Quant. Inform. 4 (2006) 1023-1035.
25. J. Kempe, Discrete quantum walks hit exponentially faster, in Proc. 7th Int. Workshop on Randomization and Approximation Techniques in Computer Science (2003), pp. 354-369, also appears on Probab. Theory Related Field 133 (2005) 215-235.
26. H. Krovi and T. A. Brun, Hitting time for quantum walks on the hypercube, Phys. Rev. A 74 (2006) 042334.
27. A. M. Childs, R. Cleve, E. Deotto, E. Farhi, S. Gutmann and D. A. Spielman, Exponential algorithmic speedup by quantum walk, in Proc. 35th ACM Symp. Theory of Computing (2003), pp. 59-68.
28. T. A. Brun, H. A. Carteret and A. Ambainis, Quantum walk with decoherent coins, Phys. Rev. A 67 (2003) 032304.
29. K. Zhang, The limiting distribution of decoherent quantum random walks, Phys. Rev. A 77 (2008) 062302.
30. T. Miyazaki, M. Katori and N. Konno, Wigner formula of rotation matrices and quantum walks, Phys. Rev. A 76 (2007) 012332.
31. H. Georgi, Lie Algebras in Particle Physics, 2nd edn. (Perseus Books, Reading, 1999).
32. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000).