7 June 2007 A systems approach to the quantum random walk
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Proceedings Volume 6603, Noise and Fluctuations in Photonics, Quantum Optics, and Communications; 66030V (2007); doi: 10.1117/12.731359
Event: SPIE Fourth International Symposium on Fluctuations and Noise, 2007, Florence, Italy
Abstract
In this paper we address the quantum random walk on the real line. Specifically, we utilize a dynamical system formulation of the walk, which leads to a momentum space expression of the probability amplitude that is a function of only the initial condition. Our focus is, for the most part, limited to the Hadamard walk. As such, this closed form expression is not as general as that given by [1]. This lack of generality is offset by the ease with which we obtain the expression, and the insight offered by it. Our closed form expression allows us to easily compute the walk pdf, hence the cdf (cumulative distribution function). It is shown that the cdf converges to its limiting form relatively quickly. We push the simple mathematics in an to attempt to obtain a closed form expression for this form. But it becomes too involved to take it to completion in this paper, without running the risk of losing appreciation for the simplicity of our approach.
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Peter Sherman, Ian Fuss, Lang White, Sanjeev Naguleswaran, "A systems approach to the quantum random walk", Proc. SPIE 6603, Noise and Fluctuations in Photonics, Quantum Optics, and Communications, 66030V (7 June 2007); doi: 10.1117/12.731359; https://doi.org/10.1117/12.731359
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KEYWORDS
Mathematics

Dynamical systems

Fourier transforms

Astatine

Optical communications

Photonics

Quantum communications

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