On Algorithms, Separability and Cellular Automata in Quantum Computing
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Date
2007-09-26T18:33:14Z
Authors
Cheung, Donny
Journal Title
Journal ISSN
Volume Title
Publisher
University of Waterloo
Abstract
In Part I of this thesis, we present a new model of quantum cellular automata
(QCA) based on local unitary operations. We will describe a set of desirable
properties for any QCA model, and show that all of these properties are
satisfied by the new model, while previous models of QCA do not. We will also
show that the computation model based on Local Unitary QCA is equivalent to
the Quantum Circuit model of computation, and give a number of applications of
this new model of QCA. We also present a physical model of classical CA, on
which the Local Unitary QCA model is based, and Coloured QCA, which is an
alternative to the Local Unitary QCA model that can be used as the basis for
implementing QCA in actual physical systems.
In Part II, we explore the quantum separability problem, where we are given a
density matrix for a state over two quantum systems, and we are to determine
whether the state is separable with respect to these systems. We also look at
the converse problem of finding an entanglement witness, which is an
observable operator which can give a verification that a particular quantum
state is indeed entangled. Although the combined problem is known to be
NP-hard in general, it reduces to a convex optimization problem, and by
exploiting specific properties of the set of separable states, we introduce a
classical algorithm for solving this problem based on an Interior Point
Algorithm introduced by Atkinson and Vaidya in 1995.
In Part III, we explore the use of a low-depth AQFT (approximate quantum
Fourier transform) in quantum phase estimation. It has been shown previously
that the logarithmic-depth AQFT is as effective as the full QFT for the
purposes of phase estimation. However, with sub-logarithmic depth, the phase
estimation algorithm no longer works directly. In this case, results of the
phase estimation algorithm need classical post-processing in order to retrieve
the desired phase information. A generic technique such as the method of
maximum likelihood can be used in order to recover the original phase.
Unfortunately, working with the likelihood function analytically is
intractable for the phase estimation algorithm. We develop some computational
techniques to handle likelihood functions that occur in phase estimation
algorithms. These computational techniques may potentially aid in the analysis
of certain likelihood functions.