Abstract
Quantum walks play an important role in the development of quantum algorithms. Algorithms based on quantum walks generally use a technique called amplitude amplification, which was introduced in Grover’s algorithm. This technique differs from the ones used in algebraic algorithms, in which the Fourier transform plays the main role. However, it is possible to go beyond Grover’s algorithm in terms of efficiency. The best algorithm to solve the element distinctness problem is based on quantum walks. This problem consists in determining whether there are repeated elements in a set of elements. When Grover’s algorithm is used, the solution is less efficient.
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References
Aharonov, D., Ambainis, A., Kempe, J., Vazirani, U.: Quantum walks on graphs. In: Proceedings of 33th STOC, pp. 50–59. ACM, New York (2001)
Aharonov, Y., Davidovich, L., Zagury, N.: Quantum random walks. Phys. Rev. A 48(2), 1687–1690 (1993)
Ambainis, A., Bach, E., Nayak, A., Vishwanath, A., Watrous, J.: One-dimensional quantum walks. In: Proceedings of 33th STOC, pp. 60–69. ACM, New York (2001)
Ambainis, A.: Quantum walk algorithm for element distinctness. In: FOCS ’04: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, pp. 22–31. IEEE Computer Society, Washington, DC (2004)
Childs, A.: On the relationship between continuous- and discrete-time quantum walk. Commun. Math. Phys. 294, 581–603 (2010)
Childs, A.M.: Universal computation by quantum walk. Phys. Rev. Lett. 102, 180501 (2009)
Childs, A.M., Farhi, E., Gutmann, S.: An example of the difference between quantum and classical random walks. Quant. Informa. Process. 1(1), 35–43 (2002)
Cover, T.M., Thomas, J.: Elements of Information Theory. Wiley, New York (1991)
Farhi, E., Gutmann, S.: Quantum computation and decision trees. Phys. Rev. A 58, 915–928 (1998)
Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 1, 3rd edn. Wiley, New York (1968)
Gould, H.W.: Combinatorial Identities. Morgantown Printing and Binding Co., Morgantown (1972)
Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics: A Foundation for Computer Science, 2nd edn. Addison-Wesley Professional, Reading (1994)
Hughes, B.D.: Random Walks and Random Environments: Random Walks (Vol 1). Clarendon Press, Oxford (1995)
Hughes, B.D.: Random Walks and Random Environments: Random Environments (Vol 2). Oxford University Press, Oxford (1996)
Kempe, J.: Quantum random walks – an introductory overview. Contemp. Phys. 44(4), 302–327 (2003) quant-ph/0303081.
Lovett, N.B., Cooper, S., Everitt, M., Trevers, M., Kendon, V.: Universal quantum computation using the discrete-time quantum walk. Phys. Rev. A 81, 042330 (2010)
Marquezino, F.L., Portugal, R.: The QWalk simulator of quantum walks. Comput. Phys. Commun. 179(5), 359–369 (2008), arXiv:0803.3459
Moore, C., Mertens, S.: The Nature of Computation. Oxford University Press, New York (2011)
Nayak, A., Vishwanath, A.: Quantum walk on a line. DIMACS Technical Report 2000-43, quant-ph/0010117 (2000)
Strauch, F.W.: Connecting the discrete- and continuous-time quantum walks. Phys. Rev. A 74(3), 030301 (2006)
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Portugal, R. (2013). Introduction to Quantum Walks. In: Quantum Walks and Search Algorithms. Quantum Science and Technology. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6336-8_3
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