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Quantum information and quantum computation
Quantum information and quantum
computation is a rapidly evolving interdisciplinary field, which
has attracted researchers from physics, computer science,
mathematics, as well as electronic engineering. Instead of
brute-force minitrization of basic electronic components, quantum
computation utilises entirely new design architecture and
promises to solve problems that are intractable on conventional
computers. At the heart of a quantum computer are the quantum
bits and quantum logic gates, which can be precisely controlled
by the laws of quantum mechanics to perform computational tasks
at remarkably fast speed. In this project, we aim to examine in
detail the operation process of several different types of
quantum logic gates. We are also interested in several other
aspects of this field, including (1) quantum cryptography and bit
commitment; and (2) quantum state entanglement and
discrimination. 
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Graph identification by quantum walks
Quantum random walks display remarkably different properties
from their classical counterparts, most notably their fast
spreading characteristics. For example, they were proven to
provide an exponential algorithmic speedup for traversing a
randomized glued-tree graph. However, despite such potentially
superior efficiency in quantum random walks, they have yet to
be applied to problems of practical importance. Graph
isomorphism is a long-standing open problem in mathematics,
which is to decide whether two given structures are
topologically identical. It also has applications in many areas
of science and engineering. For instance, it is often
critically important in chemistry and molecular biology that we
know if two molecules have topologically the same structure, a
generalization of which is a graph with specified nodes and
connectivity. Graph isomorphism identification provides an
efficient tool for protein structure comparison and
classification. It can also be used for structural analysis of
kinematic chains, oil pipelines, roads and subways, scheduling
problems, network management, communication systems, pattern
recognition, data management and retrieval, logic design and
switching theory. In this project, we aim to develop and prove
an algorithm based on quantum walks to solve this important
problem.  
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Physical Implementation of Quantum Random Walks
This project addresses key issues
in the actual physical implementation of quantum random walks in
the laboratory. In particular, we need a nano system that (1)
exhibits discretized and addressable states representing the
nodes and edges of graphs, (2) allows selective mixing of states
to implement the coin superposition operator, (3) allows
controlled quantum evolution to implement the conditional shift
operator, and (4) permits accurate measurement of its quantum
states. There have been several proposals using a variety of
solid state as well as optical schemes. A detailed investigation
will be carried out to examine the merits and shortcomings of
these proposals.  
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Electronic structure of artificial atoms (quantum dots)
Quantum dots are artificially fabricated "atoms", in which
charge carriers are confined in all three dimensions just like
electrons in real atoms. Consequently, they exhibit similar
properties normally associated with real atoms, such as quantised
energy levels and shell structures. What makes these artificial
atoms special is that their sizes and shapes can be precisely
controlled by advanced nanofabrication technology to achieve
user-designed atomic properties. This has opened up a wide range
of possibilities and areas for exploring new physics and new
applications. For example, quantum dots may be used to build
lasers with otherwise inaccessible wavelengths. They may become
the building blocks for the next-generation computer chips. It is
also hoped that quantum dots may one day be used to help
realising the dream of quantum computation. Our current focus is
to examine the electronic structure of quantum dots of various
size and shape, as well as the associated scattering processes
from these quantum dots. 
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Quantum simulation using nano-meter-scale systems
Richard Feynman was among the first to note that simulating
quantum dynamics using a classical computer was intrinsically
hard due to the exponentially increasing Hilbert space occupied
by a linearly increasing number of particles in the system.
However, Feynman suggested that it was possible to construct a
“quantum simulator”, which could be programmed to
simulate the behavior of any quantum system of interest. For
instance, a fairly complicated molecule can be modeled
accurately by a quantum simulator with a small number of
well-controlled qubits. To build such a quantum simulator would
be a tremendous step forward towards the eventual goal of
achieving a general universal quantum computer. The aim of this
project is to engineer and control the properties of quantum
dot systems to study the electronic structure of molecular
systems. 
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Quantum dynamics of qubits and qugates
Quantum computing is a process in
which a register of qubits (the quantum analogue of classical
bits) is evolved from some initial state to a final state through
a series of logic gate operations, which obey the laws of quantum
mechanics. From the measurement of this final state we obtain the
computation results. Quantum computing exploits the nature of the
quantum world in a way that promises to solve problems that are
intractable on conventional computers. Its deep connection to
information theory has also prompted a revision of our previous
understanding of the physics of information. Essential to quantum
computing is the physical realization of qubits and quantum logic
manipulations, which can be precisely controlled by the laws of
quantum mechanics to perform computational tasks at remarkably
fast speed. In this project, we aim to examine in detail the
operation process of a complete set of quantum logic gates
implemented in nano-meter-scaled systems.  
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Nanostructured electronic devices
As electronic circuits get progressingly smaller down to the nanometer scale, device analysis based on classical or semi-classical transport theories no longer works since the quantum wave nature of the electrons starts to play a dominant role. Contemporary advances in semi-conductor fabrication technology have already allowed construction of nanostructured devices from 1nm to 100nm in size and confined in two, one and zero dimensions. Since the wavelength of electrons with energy of a few meV is around tens of nano meters, the quantum wave nature of the electrons starts to play a dominant role in the functionality of electronic devices.
One of the possibilities is for the devices to operate by controlling the phase of quantum electron waves rather than the electron density as in present-day devices. This opens an entirely new horizon in computer electronics. For example, the source-to-drain current can be switched on and off in response to extremely small changes in the charge on the gate amounting to one electron or less. Consequently, it offers the potential of ultra-high speed operation at very-low-power levels. Another possibility is to establish communications between quantum wires and quantum dot cells through the nonlocality nature of the electron waves without the need of traditional wires to propagate information. This provides new powerful architectures for nano-computer logic designs, such as the transistorless quantum-dot cellular automata paradigm. This project aims to provide an accurate description of quantum phenomena in these basic components.
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Scattering Theory
The principal objective is to
develop a highly accurate time-dependent approach to quantum
scattering, including (1) electron/positron collision with
individual atoms/ions and (2) electron transport in quantum
cavities. This approach will have many distinct advantages over
the traditional time-independent methods. For example, it
provides information on transient behaviours and allows direct
visualisation of the collision or transport process, where one
can "watch" the electron and atom wavepacket evolving in time and
in space; it handles continuum states, ionisation, resonance and
post-collision interaction in the most natural way and is thus
free of the inherited difficulties encountered by the
time-independent approaches; as an initial value problem, it is
also comparatively easy to implement, flexible, and versatile in
treating a large variety of quantum many-body problems. It is
therefore expected to (1) resolve some serious discrepancies
between existing theories and experiments, (2) reveal new
physical phenomena otherwise hidden, and (3) identify and
characterise impurities in mesoscopic structures.
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Density matrix theory on complete scattering experiments
Collaboration with Prof. J.F. Williams and Prof. A.T.
Stelbovics
The scattering of electrons from
atoms occur in a great variety of environment such as in plasma
and atmosphere. Unfortunately, the complexity of the natural
environment prevents a direct quantitative analysis of the
collision processes. For this purpose, well controlled scattering
experiments are carried out in laboratories. The main theme of
this project is to provide theoretical guidance for optimum
designs of such experiments, with the aim of obtaining complete
quantum mechanical available information on several basic
scattering events. Such information will provide the most
stringent test of theoretical models for treating quantum
mechanical many-particle systems. Once the validity of certain
theoretical model is established, reliable calculations of
transition energies, collision cross sections, alignment and
orientation parameters and other atomic data can be performed.
These data are essential but sometimes inaccessible otherwise for
describing the physics of plasma and fusion, laser and discharge
systems, planetary atmospheres, as well as astronomical
events.
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Acoustic wave propagator and its applications
Collaboration with Prof. J. Pan
The aim of this project is to
extend the quantum wave propagator approach to study sound wave
propagation in air and in solids. This method has the potential
to give solutions to a range of long-standing acoustic problems,
such as transient decay of acoustic waves in an enclosure; sound
wave propagation in inhomogeneous and non-linear media; acoustic
wave variation in spaces with moving boundaries. These problems
are extremely difficult, if not intractable, to be treated by
conventional frequency domain methods such as FEM and BEM.
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