QUANTUM WALKS

Acta Physica Slovaca

Volumn 61, Issue 6, 2011, Pages 603-725

  Reitzner, Daniel ^a   Nagaj, Daniel ^b   Bužek, Vladimír ^b  

^a TECHNICAL UNIVERSITY OF MUNICH   (Germany)

^b INSTITUTE OF PHYSICS   (Slovakia)

Author keywords

Markov Chains; Quantum Algorithms; Quantum Walks; Random Walks

Indexed keywords

EID: 85195098307     PISSN: 03230465     EISSN: 1336040X     Source Type: Journal    
DOI: 10.2478/v10155-011-0006-6     Document Type: Article

References (97)

1. F. Galton: Natural Inheritance (The Macmillan Company, London, 1889).
2. L. Bachelier: Theorie de la speculation (PhD thesis, Annales Scientifiques de l’Ecole Superieure III-17, 21, 1900).
3. N. Wiener: Differential space, J. Math. Phys. 2, 131 (1923).
4. A. Einstein: Über die von der molekularkinetischen Theorie der wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen, Ann. Physik 17, 549 (1905).
5. R. Brown: A brief account of microscopical observations made in the months of June, July and August, 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies, Phil. Mag. 4, 161 (1928).
6. J. Perrin: Mouvement brownien et réalité moléculaire, Ann. Chim. Phys. 18, 5 (1909).
7. B.D. Hughes: Random Walks and Random Environments, Clarendon Press (1995)
8. A. Sinclair: Algorithms for Random Generation and Counting, a Markov Chain Approach, Birkhauser (1993).
9. Rajeev Motwani and Prabhakar Raghavan, Randomized algorithms, ACM Comput. Surv. 28, 1 33–37 (March 1996).
10. N. Metropolis, A.W. Rosenbluth, N.N. Rosenbluth, A.H. Teller, E. Teller: Equation of State Calculations for Fast Computing Machines, J. Chem. Phys. 21, 1087 (1953).
11. W.K. Hastings: Monte Carlo Sampling Methods Using Markov Chains and Their Applications, Biometrika 57, 97 (1970).
12. S. Kirkpatrick, C.D. Gelatt Jr., M.P. Vecchi: Optimization by Simulated Annealing, Science 220, 671 (1983).
13. J. Černý: Thermodynamical Approach to the Travelling Salesman Problem, J. Opt. Theory Appl. 45, 41 (1985).
14. Y. Aharonov, L. Davidovich, N. Zagury: Quantum random walks, Phys. Rev. A 48, 1687 (1993).
15. D. Aharonov, A. Ambainis, J. Kempe, U. Vazirani: Quantum walks on graphs, in STOC’01: Proceedings of the thirty-third annual ACM symposium on Theory of computing, 50–59 (New York, NY, USA, ACM, 2001).
16. A. Ambainis, E. Bach, A. Nayak, A. Vishwanath, and J. Watrous: One-dimensional quantum walks, in STOC’01: Proceedings of the thirty-third annual ACM symposium on Theory of computing, 37–49 (New York, NY, USA, ACM, 2001).
17. V. Kendon: Decoherence in quantum walks — a review, Mathematical Structures in Computer Science 17, 1169 (2007).
18. V. Kendon, B. Tregenna: Decoherence can be useful in quantum walks, Phys. Rev. A 67, 042315 (2003).
19. T.A. Brun, H.A. Carteret, A. Ambainis: Quantum random walks with decoherent coins, Phys. Rev. A 67, 032304 (2003).
20. J. Košík, V. Bužek, and M. Hillery: Quantum walks with random phase shifts, Phys. Rev. A 74, 022310 (2006).
21. M. Hillery, J. Bergou, E. Feldman: Quantum walks based on an interferometric analogy, Phys. Rev. A 68, 032314 (2003).
22. F.M. Andrade, M.G.E. da Luz: Equivalence between discrete quantum walk models in arbitrary topologies, Phys. Rev. A 80, 052301 (2009).
23. N. Shenvi, J. Kempe, K.B. Whaley: Quantum random-walk search algorithm, Phys. Rev. A 67, 052307 (2003).
24. V. Kendon: Quantum Walks on General Graphs, Int. J. Quantum Info. 4, 791 (2006).
25. S. Severini: On the digraph of a unitary matrix, SIAM J. Matrix Anal. Appl. (SIMAX) 25, 295 (2003).
26. A. Montanaro: Quantum walks on directed graphs, Quantum Information and Computation 7, 93 (2007).
27. B. Tregenna, W. Flanagan, R. Maile, V. Kendon: Controlling discrete quantum walks: coins and initial states, New Journal of Physics 5, 83 (2003).
28. Y. Omar, N. Paunkovic, L. Sheridan, S. Bose: Quantum walk on a line with two entangled particles, Phys. Rev. A 74, 042304 (2006).
29. C. Moore, A. Russell: Quantum walks on the hypercube, in RANDOM, 164–178 (2002).
30. L.K. Grover: Quantum mechanics helps in searching for a needle in a haystack, Phys. Rev. Lett. 79, 325 (1997).
31. J. Kempe: Discrete quantum walks hit exponentially faster, Probability Theory and Related Fields 133, 215 (2005).
32. A. M. Childs, R. Kothari, Simulating sparse Hamiltonians with star decompositions, In proceedings of TQC 2010, Lecture Notes in Computer Science 6519, pp. 94-103 (2011).
33. A.M. Childs, R. Cleve, E. Deotto, E. Farhi, S. Gutmann, D.A. Spielman, Exponential algorithmic speedup by quantum walk, Proceedings of the 35th ACM Symposium on Theory of Computing, pp. 59–68 (2003).
34. E. Bach, S. Coppersmith, M.P. Goldschen, R. Joynt, J. Watrous: One-dimensional quantum walks with absorbing boundaries, Journal of Computer and System Sciences 69, 562 (2004).
35. T. Yamasaki, H. Kobayashi, H. Imai: Analysis of absorbing times of quantum walks, Phys. Rev. A 68, 012302 (2003).
36. L.K. Grover: Quantum Computers Can Search Rapidly by Using Almost Any Transformation, Phys. Rev. Lett. 80, 4329 (1998).
37. G. Brassard, P. Høyer, M. Mosca, A. Tapp: Quantum Amplitude Amplification and Estimation, preprint arXiv:quant-ph/0005055.
38. A. Ambainis: Quantum walk algorithm for element distinctness, SIAM J. Comp. 37, 210 (2007).
39. A.M. Childs, J.M. Eisenberg: Quantum algorithms for subset finding, Quantum Information and Computation 5, 593 (2005).
40. F. Magniez, M. Santha, M. Szegedy: Quantum Algorithms for the Triangle Problem, in SODA’05: Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms, 1109 (Philadelphia, PA, USA, 2005).
41. H. Buhrman, R. Špalek: Quantum Verification of Matrix Products, in SODA’06: Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm, 880 (New York, NY, USA, 2006).
42. F. Magniez, A. Nayak: Quantum Complexity of Testing Group Comutativity, Algorithmica 48, 221 (2007).
43. A. Ambainis, J. Kempe, A. Rivosh: Coins make quantum walks faster, in SODA’05: Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms, 1099 (Philadelphia, PA, USA, 2005).
44. A. M. Childs, J. Goldstone, Spatial search by quantum walk, Physical Review A 70, 022314 (2004).
45. D. Reitzner, M. Hillery, E. Feldman, V. Bužek: Quantum Searches on Highly Symmetric Graphs, Phys. Rev. A 79, 012323 (2009).
46. V. Potoček, A. Gábris, T. Kiss, I. Jex: Optimized quantum random-walk search algorithms on the hypercube, Phys. Rev. A, 79, 012325 (2009).
47. M. Boyer, G. Brassard, P. Høyer, A. Tapp: Tight bounds on quantum searching, Fortschritte der Physik 46, 493 (1998).
48. S.L. Braunstein, V. Bužek, M. Hillery: Quantum-information distributors: Quantum network for symmetric and asymmetric cloning in arbitrary dimension and continuous limit, Phys. Rev. A 63, 052313 (2001).
49. E. Feldman, M. Hillery, H.-W. Lee, D. Reitzner, H. Zheng, V. Bužek: Finding structural anomalies in graphs by means of quantum walks, Phys. Rev. A 82, 040301(R) (2010).
50. A. Tulsi: Faster quantum-walk algorithm for the two-dimensional spatial search, Phys. Rev. A 78, 012310 (2008).
51. M. Szegedy, Quantum Speed-up of Markov Chain Based Algorithms, Proc. of 45th Annual IEEE Symposium on Foundations of Computer Science, pp. 32–41 (2004).
52. M. Santha, Quantum Walk Based Search Algorithms, Proc. of 5th Theory and Applications of Models of Computation (TAMC08), Lectures Notes on Computer Science, vol. 4978, pp. 31–46, 2008.
53. L. K. Grover A Different Kind of Quantum Search, arXiv:quant-ph/0503205.
54. T. Tulsi, L. K. Grover, and A. Patel, A new algorithm for fixed point quantum search, Quantum Info. Comput. 6 (6), 483–494 (2006).
55. G. Brassard, P. Høyer, and A. Tapp, Quantum Counting, Proc. of 25th International Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science, vol. 1443, pp. 820–831, 1998. quant-ph/9805082.
56. K. Temme, T. J. Osborne, K. G. Vollbrecht, D. Poulin, F. Verstraete, Quantum Metropolis sampling, Nature 471, 87–90, http://arXiv:0911.3635.
57. M.-H.Yung, A. Aspuru-Guzik, A Quantum-Quantum Metropolis Algorithm, arXiv:1011.1468.
58. C. Marriott and J.Watrous, Quantum Arthur-Merlin games, Computational Complexity, 14 (2) 122–152 (2005).
59. D. Nagaj, P. Wocjan, Y. Zhang, Fast QMA Amplification, QIC Vol. 9 No.11&12 (2009), 1053-1068 [arXiv:0904.1549].
60. R. Somma, S. Boixo, H. Barnum, E. Knill, Quantum Simulations of Classical Annealing Processes, arXiv:0804.1571.
61. 4) Volume Algorithm, Journal of Computer and System Sciences, vol. 72, issue 2, pp. 392–417, 2006.
62. M. Jerrum, A. Sinclair, and E. Vigoda, A Polynomial-Time Approximation Algorithm for the Permanent of a Matrix Non-Negative Entries, Journal of the ACM, vol. 51, issue 4, pp. 671–697, 2004.
63. M. Jerrum and A. Sinclair, Polynomial-Time Approximation Algorithms for the Ising Model, SIAM Journal on Computing, vol. 22, pp. 1087–1116, 1993.
64. I. Bezáková, D. Štefankovič, V. Vazirani and E. Vigoda, Accelerating Simulated Annealing for the Permanent and Combinatorial Counting Problems, SIAM Journal on Computing, vol. 37, No. 5, pp. 1429–1454, 2008.
65. D. Aldous, J. London Math. Soc. 253, 564 (1982).
66. M. Jerrum, L. Valiant and V. Vazirani, Random Generation of Combinatorial Structures from a Uniform Distribution, Theoretical Computer Science, vol. 43, issue 2-3, pp. 169–188, 1986.
67. D. Štefankovič, S. Vempala, and E. Vigoda, Adaptive Simulated Annealing: A Near-Optimal Connection between Sampling and Counting, Proc. of the 48th Annual IEEE Symposium on Foundations of Computer Science, pp. 183-193, 2007.
68. P. Wocjan and A. Abeyesinghe, Speed-up via Quantum Sampling, Physical Review A, vol. 78, pp. 042336, 2008.
69. P. Wocjan et.al., Quantum Algorithm for Approximating Partition Functions, Physical Review A 80, 022340 (2009).
70. A. Childs, Quantum Algorithms, lecture notes, University of Waterloo, http://www.math.uwaterloo.ca/ amchilds/teaching/w11/qic823.html (2011).
71. M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000.
72. P. Richter, Quantum Speed-Up of Classical Mixing Processes, Physical Review A, vol. 76, 042306, 2007.
73. R. Somma, S. Boixo, and H. Barnum, Quantum Simulated Annealing, arXiv:0712.1008.
74. C. H. Bennett, E. Bernstein, G. Brassard, and U. Vazirani, Strengths and Weaknesses of Quantum Computing, SIAM J. Comput. 26, 5 (October 1997), 1510-1523.
75. E. Farhi, S. Gutmann, Analog Analogue of Digital Quantum Computation, Phys. Rev. A 57, 2403-2406 (1998).
76. E. Farhi, J. Goldstone, S. Gutmann, M. Sipser, Quantum computation by adiabatic evolution, quant-ph/0001106 (2000).
77. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, Chapter 6.5, Cambridge University Press (1992).
78. E. Farhi, S. Gutmann, Quantum Computation and Decision Trees, Phys. Rev. A 58 (2), 915 (1998).
79. A. M. Childs, E. Farhi, and S. Gutmann, An example of the difference between quantum and classical random walks, Quantum Information Processing 1, 35–43 (2002).
80. A. M. Childs, J. Goldstone, Spatial search and the Dirac equation, Physical Review A 70, 042312 (2004).
81. A. M. Childs, E. Deotto, E. Farhi, J. Goldstone, S. Gutmann, A. J. Landahl, Quantum search by measurement, Physical Review A 66, 032314 (2002).
82. A. M. Childs, Quantum Information Processing in Continuous Time, Ph.D. Thesis, Massachusetts Institute of Technology (2004).
83. E. Farhi, J. Goldstone, S. Gutmann, A quantum algorithm for the Hamiltonian NAND tree, Theory of Computing, Vol. 4, no. 1, pp.169-190, 2007; quant-ph/0702144.
84. A. M. Childs, R. Cleve, S. P. Jordan, D. L. Yonge-Mallo, Discrete-query quantum algorithm for NAND trees, Theory of Computing, vol. 5, no. 1, pp. 119-123, 2009.
85. (1) on a quantum computer, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS’07), 2007.
86. P. Hoyer, R. Špalek, Lower bounds on Quantum Query Complexity, Bulletin of the EATCS 87, 2005; quant-ph/0509153.
87. D. Nagaj, R. D. Somma, M. Kieferová, Quantum Speedup by Quantum Annealing arXiv:1202.6257 (2012).
88. D. Nagaj, Universal 2-local Hamiltonian Quantum Computing, Physical Review A 85, 032330 (2012).
89. D. Nagaj, Local Hamiltonians in Quantum Computation, Ph.D. thesis, MIT (2008), arXiv:0808.2117.
90. M. Kieferová, D. Nagaj, Quantum Walks on Necklaces and Mixing, International Journal of Quantum Information Vol. 10, Issue 2, 1250025 (2012).
91. A. M. Childs, Universal computation by quantum walk Physical Review Letters 102, 180501 (2009).
92. A. M. Childs, D. Gosset, Z. Webb, Universal computation by multi-particle quantum walk arXiv:1205.3782 (2012).
93. N. B. Lovett, S. Cooper, M. Everitt, M. Trevers, V. Kendon, Universal quantum computation using the discrete-time quantum walk, Physical Review A 81, 042330 (2010).
94. D. Janzing, P. Wocjan, A PromiseBQP-complete String Rewriting Problem, Quantum Information and Computation, vol. 10, no. 3&4, pp. 234-257 (2010).
95. A. Childs, On the Relationship Between Continuous- and Discrete-Time QuantumWalk, Commun. Math. Phys. 294, 581-603 (2010).
96. G. Brassard, Searching a quantum phone book, Science 275, 627 (1997).
97. Catalan number, Wikipedia, http://en.wikipedia.org/wiki/Catalan number (2012).