We explore the application of a quantum algorithm to optimisation problems over a structured space. For
example, problems in automated planning can be represented as automata. These automata are shown to posses
algebraic structure that can be exploited by a quantum period finding algorithm. The fact that the quantum
walk also provides exponential speed-up over these same structures is of particular interest and results of our
investigation will be presented.
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 [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.
There has been recent interest in implementing automated planning by optimizing a planning domain modeled as a stochastic system. Planning is viewed as a process where sequential decision problems are solved
in order to reach the goal, and thus, can be considered as instances of a Markov Decision Process (MDP). However, standard MDP techniques cannot solve a typical planning problem in polynomial time. Hence, the
motivation for investigating the use of quantum search techniques based on the Grover Search Algorithm, to identify policies with high utility.
This paper describes a numerical design procedure for the design of
optimal (in the minimum mean square channel estimation error sense)
precoders for diversity systems with unitary codes where temporal
correlation in the channel is present. The paper describes the use
matrix differential calculus to derive the Jacobian (first derivative)
of trace of the optimal (MMSE) estimation error covariance matrix with respect to the channel precoder matrix. This is used to define a gradient descent algorithm to minimise this error over all unit norm
precoders. Simulation results using a per-survivor processing based
receiver illustrate the improvement gained over unitary codes. The
paper also shows by example that this does not necessarily lead
on its own to optimal symbol detection error probability performance.
This paper describes the application of the theory of projections onto convex sets to time-frequency filtering and synthesis problems. We show that the class of Wigner-Ville Distributions (WVD) of L2 signals form the boundary of a closed convex subset of L2(R2). This result is obtained by considering the convex set of states on the Heisenberg group of which the ambiguity functions form the extreme points. The form of the projection onto the set of WVDs is deduced. Various linear and non-linear filtering operations are incorporated by formulation as convex projections. An example algorithm for simultaneous time-frequency filtering and synthesis is suggested.
KEYWORDS: Radar, Chemical elements, Wavelets, Silicon, Defense and security, Signal analysis, Matrices, Wavelet transforms, Electronics, Analytical research
This paper addresses the problem of designing signals for general group representations subject to constraints which are formulated as convex sets in the Hilbert space of the group states. In particular, the paper considers irreducible representations in an infinite dimensional Hilbert space and derives an iterative producedure for proceeding from an arbitrary element of the Hilbert space to a state of the group subject to a priori imposed constraints with closed convex range. As examples, the paper focuses on narrowband and wideband radar ambiguity synthesis.
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