Quantum Walks for Computer Scientists (Synthesis Lectures on Quantum Computing)

Quantum Walks for Computer Scientists (Synthesis Lectures on Quantum Computing)

Language: English

Pages: 134

ISBN: 1598296566

Format: PDF / Kindle (mobi) / ePub


Quantum computation, one of the latest joint ventures between physics and the theory of computation, is a scientific field whose main goals include the development of hardware and algorithms based on the quantum mechanical properties of those physical systems used to implement such algorithms. Solving difficult tasks (for example, the Satisfiability Problem and other NP-complete problems) requires the development of sophisticated algorithms, many ofwhich employ stochastic processes as their mathematical basis. Discrete random walks are a popular choice among those stochastic processes. Inspired on the success of discrete random walks in algorithm development, quantum walks, an emerging field of quantum computation, is a generalization of random walks into the quantum mechanical world. The purpose of this lecture is to provide a concise yet comprehensive introduction to quantum walks. Table of Contents: Introduction / Quantum Mechanics / Theory of Computation / Classical Random Walks / Quantum Walks / Computer Science and Quantum Walks / Conclusions

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Feynman, in his traditional and celebrated style, lectured at MIT [5] about the fundamental capabilities and limitations of classical computers to simulate quantum systems. A gentle and concise introduction to this blend of physics, computer science and information theory, as well as Feynman’s main ideas behind physics and computation can be found in [26]. In 1985, Deutsch made two key contributions in [66]: a design of a Universal Quantum Turing Machine, and a physics-oriented version of the

unrestricted DQWL, it was shown in [126] that σ (X) t √ 2−1 → as t → ∞. In any case, the standard deviation of the unrestricted Hadamard 2 DQWL is O(t) and that result is in contrast with the standard deviation of an unrestricted √ classical random walk on a line, which is O( t) (Eq. (4.12)). 3. Mixing time. It was shown in [96, 113] that an unrestricted Hadamard DQWL has a linear (q ) (q ) mixing time τ = O(t), where t is the number of steps. Furthermore, τ was compared with the corresponding

with |V | = n then a continuous time random walk on G can be described by the order n infinitesimal generator matrix M given by    −γ , a = b, (a, b) ∈ G Mab = 0,   kγ , a = b, (a, b) ∈ /G (5.37) a = b and k is the valence of vertex a. Following [167, 182], the probability of being at vertex a at time t is given by d p a (t) =− dt Mab p b (t). (5.38) b Now, let us define a Hamiltonian [167, 182] that closely follows Eq. (5.37). Definition 5.4.2. Let Hˆ be a Hamiltonian with matrix

Brown, The Quest for the Quantum Computer. New York: Touchstone, 2001. [2] J. Volpi, In Search of Klingsor. London: Fourth Estate Ltd, 2004. [3] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information. Cambridge: Cambridge University Press, 2000. [4] A. Y. Kitaev, A. H. Shen, and M. N. Vyhalyi, Classical and Quantum Computation. (Graduate Studies in Mathematics 47), Providence, RI: American Mathematical Society, 1999. [5] R. P. Feynman, “Simulating physics with computers,”

quantum walk,” Phys. Rev. A, Vol. 76, p. 022316, 2007. doi:10.1103/PhysRevA.76.022316 [143] V. Kendon and B. Tregenna, “Decoherence can be useful in quantum walks,” Phys. Rev. A, Vol. 67, p. 042315, 2003. doi:10.1103/PhysRevA.67.042315 [144] V. Kendon and B. Tregenna, “Decoherence in a quantum walk on the line,” in Proc. QCMC, 2002. [145] E. Bach, S. Coppersmith, M. Paz Goldshen, R. Joynt, and J. Watrous, “Onedimensional quantum walks with absorbing boundaries,” J. Comput. Syst. Sci., Vol. 69(4),

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