**Brendan Douglas** — Graph isomorphism problem and quantum algrorithms

**Matthew Katz** (Master of Science and Technology) — Graph-state quantum computation

**Kia Manouchehri** — Quantum Random Walks

**Bradley McGrath** — Acoustic Wave Trapping Barriers

**Bahaa Raffah** — Quantum dynamics and structure of few-electron systems

**Filip Welna** — Heterogeneous cluster methods for quantum dynamical simulations

**Hongmei Sun ^{#}** — Acoustic wave propagator

**Chris Dinneen** — Theoretical modelling of quantum logic gates **
Chris Hines** (PhD) — Electronic Properties of Quantum dots

Peter Falloon

Chris Carter

Quantum information and computation is a rapidly evolving interdisciplinary field that has attracted researchers from physics, computer science, mathematics, chemistry, and electronic engineering.

Instead of brute-force miniaturisation of basic electronic components, quantum computation uses entirely new design architecture and promises to solve problems that are intractable on conventional computers.

Quantum information offers the prospect of harnessing nature at a much deeper level than ever before, as well as a wealth of new possibilities for communication and data processing.

- Quantum information and quantum computation
- Quantum Graph identification by quantum walks
- Physical implementation of quantum random walks
- Electronic structure of artificial atoms (quantum dots)
- Quantum simulation using nano-meter-scale systems
- Quantum dynamics of qubits and qugates
- Nanostructured electronic devices
- Scattering in atomic and mesoscopic systems
- Density matrix theory on complete scattering experiments
- Acoustic wave propagator and its applications
- Visualisation of atomic and molecular systems
- Applications of Computer Algebra Systems (CAS)
- Linking CAS to standard numerical libraries
- Efficient computation of angular momentum coupling coefficients using CAS

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.

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.

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.

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.

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.

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.

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.

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.

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.