The first general analytic solutions for the one-dimensional quantum walk in position and momentum space are derived. These solutions reveal new symmetry features of quantum walk probability densities and insight into the behaviour of their moments. The analytic expressions for the quantum walk probability distributions provide a means of modelling quantum phenomena that is analogous to that provided by random walks in the classical domain.
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 . 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.
The statistical problem of estimating the bandwidth parameter of a Gauss-Markov process from a realization of fixed and finite duration T at selectable sampling interval Δ is addressed in this paper. As the observation time, T, is fixed and finite, the variance of estimated autocorrelation and continuous-time parameter does not
vanish as Δ approaches 0. This necessitates a second order Taylor expansion in deriving the parameter estimator bias and variance, which produces significantly more accurate bias and variance results than a first order one does. Using likelihood ratio methods, we also show that even the large sample distributions of β estimator are better modeled by a gamma than by a normal form. According to the gradient change of the variance, a key result is that three sample regions, which are termed finite, large and very large, corresponding to substantial, gradual, and very slight decrease in variance of the parameter estimator respectively, are quantified. In terms of analysis BW, the three regions are (-23,-35), (-35,-55) and (-55,-∞) dB. The characterization of the trade off between the variance decrease and sampling rate results in a practical guideline for choosing sampling rate. To demonstrate the practical value of our results, we apply them to the noise prediction problems of a time invariant GM processes. Using moment generating functions, we are able to arrive at explicit and accurate relations between, the set of variables (Δ, T, β) and m-step prediction performance. In particular, we show that prediction performance is highly robust with respect to estimation accuracy of β. This is significant, in that it allows one to use a surprisingly small observation time, T , and still achieve nearly optimal performance associated with perfect knowledge of β.
Conference Committee Involvement (2)
Noise in Communication Systems
24 May 2005 | Austin, Texas, United States
Noise in Communication
26 May 2004 | Maspalomas, Gran Canaria Island, Spain