Pre-requisite modules
Co-requisite modules
- None
Prohibited combinations
- None
Occurrence | Teaching cycle |
---|---|
A | Autumn Term 2018-19 to Spring Term 2018-19 |
Quantum Mechanics has left the laboratory to become the basis of the newest technologies. In this module we will look at some of the mathematical techniques which underlie this fascinating theory as well as discover the basis of one of the technologies directly stemming from Quantum Mechanics.
The field of Quantum Computation has been expanding exponentially over the last decade. At its core is the idea of improving computer performances to levels unreachable by standard (i.e. "classical") computer by using the very fundamentals of quantum mechanics.
The Advanced Theoretical Techniques part of this module introduces mathematical ideas and tools which are essential to modern theoretical physics. Variational principles and Lagrangian mechanics have applications in many areas of physics, and are an essential to modern classical and fundamental quantum field theories. Mathematical ideas about reference frames and coordinate systems underpin special and general relativity, and are naturally expressed using tensors. This course will teach you how to use these mathematical tools to analyse challenging theoretical physics problems from a range of areas.
Section I: Integral Transforms & Variational Methods (9 lectures)
In this part, we look at Fourier transforms (first encountered in Maths III) in more detail, and develop a related concept, the Laplace transform. We will then see some applications of these transforms, including their use in solving differential equations. We then go on to look at a more advanced form of calculus, including functional differentiation and the calculus of variations. We will conclude by looking at the ubiquity of these ideas in many different areas of physics, often expressed as some form of “variational principle”, including applications in classical mechanics, optics, field theories and quantum mechanics.
Section II: Tensors (9 lectures)
A cornerstone of modern physics is the notion that no observer is more privileged than any other in terms of being able to deduce the laws of nature. This concept is manifested in the covariant nature of our mathematical description of the universe, and the properties of the algebraic quantities (which we call tensors) which represent physical quantities. In this second section of the module we will see why some matrices can represent physical quantities and others can’t, and demonstrate how various physical laws exhibit Galilean or Lorentzian invariance. We also look at non-orthogonal coordinate systems, which are relevant to general relativity (among other applications).
Section III: Introduction to Quantum Computation (18 lectures)
The field of Quantum Computation has been expanding exponentially over the last decade. At its core is the idea of finding a physical system with the right characteristics to build the "quantum computer", a device which can improve computer performance to levels unreachable by standard (i.e. "classical") computer. There are proposals for quantum computers based on semiconductors, superconductors, cold ions or atoms, molecules in a solvent, fullerenes and so on. Each of the proposals has advantages and disadvantages, and has been partially tested experimentally. The requirements to build a quantum computer are experimentally very challenging, so that the experiments performed in this area are at the very edge of modern techniques. The "quantum computer" is in fact based on the smallest possible quantum system (the two-level system or "quantum-bit") and on exquisite quantum mechanical properties, such as state superposition.
The Introduction to Quantum Computation part of this module aims to provide an introduction to this booming research field.
Please note, students taking this module should either have taken the prerequisite modules listed above (Quantum Physics II - PHY00032I and Mathematics II - PHY00030I) or the appropriate equivalents.
Syllabus
Fundamentals of quantum computation:
concept of quantum bit (qubit);
concept of basis set
examples of physical systems used as qubits;
Bloch sphere and single qubit representation
single qubit gates
Pauli matrices
circuit representation of single qubit gates
two qubit states: Dirac and vectorial representation
two qubit gates and their matrix representation
tensor product between qubit gates and between qubit states
circuit representation of two qubit gates
role of superposition principle (quantum parallelism);
concept of entanglement; differentiating between entangled and non-entangled states
Bell states; EPR paradox and Bell inequality; significance of Bell inequality for Quantum Mechanics
Concept of teleportation; teleportation protocol for one qubit
quantum circuits
improvements of quantum over standard 'classical' computation and problem complexity
concept of density matrix and its properties; concept and differences between pure and mixed states; density matrix and decoherence
Quantum algorithms
Concept of quantum error correction; three-qubit code error correction
Requirements for physical systems to be used as quantum computers: Di Vincenzo check list
physical systems proposed as quantum computers: ion trap quantum computer, quantum-dot-based quantum computer, silicon-based NMR quantum processor, liquid state NMR quantum processor
For each proposal: how two qubit gates translate into physical interactions; main physical limitations to quantum computation (decoherence and scalability)
Experiments related to specific proposals based on semiconductor structures.
Generalities on one-way quantum computation
Integral Transforms: Motivation
Fourier transform & inverse transform
Dirac delta
Derivatives and solving differential equations
Convolution theorem
Laplace transform
Existence requirements
Techniques for finding inverse transforms
Derivatives and solving differential equations
Convolution theorem
Applications of Fourier and Laplace transforms
Variational Methods: Motivation
Functional differentiation
Calculus of variations
Extremal values & the Euler-Lagrange equation
Conservation Laws and Symmetry
Boundary conditions and constraints
Extension to vector equations and fields
Applications of Variational Methods
classical mechanics, optics, classical field theories and quantum mechanics
Tensors: Motivation
no special observers
distinction between a physical quantity and its component representation
Notation
Einstein summation notation
Kronecker delta
Levi-Civita alternator
Examples of vector identities
Examples of anisotropy, which may include the fluid stress tensor and/or the electromagnetic dielectric tensor, and their physical consequences
General coordinate transformations
Rotations and translations in two and three dimensions
What makes a tensor: concepts of invariance.
Euclidean tensors & their invariance
Minkowski space and Lorentzian invariance of four vectors, including the proper time and electromagnetic potential
Non-orthogonal coordinate systems:
motivation (i.e. why make things complicated?)
general definition of a coordinate
covariant and contravariant basis vectors
the metric tensor and its use in the evaluation of tensor quantities.
Students will be required to sit a closed examination with two sections, one for each part of the module. Students must answer two questions – one from Section A and one from Section B.
Task | Length | % of module mark |
---|---|---|
Essay/coursework Physics practice questions |
N/A | 14 |
University - closed examination Advanced Theoretical Techniques & Introduction to Quantum Computing |
3 hours | 86 |
None
Task | Length | % of module mark |
---|---|---|
University - closed examination Advanced Theoretical Techniques & Introduction to Quantum Computing |
3 hours | 86 |
Physics Practice Questions (PPQs) - You will receive the marked scripts via your pigeon holes. Feedback solutions will be provided on the VLE or by other equivalent means from your lecturer. As feedback solutions are provided, normally detailed comments will not be written on your returned work, although markers will indicate where you have lost marks or made mistakes. You should use your returned scripts in conjunction with the feedback solutions.
Exams - You will receive the marks for the individual exams from eVision. Detailed model answers will be provided on the intranet. You should discuss your performance with your supervisor.
Advice on academic progress - Individual meetings with supervisor will take place where you can discuss your academic progress in detail.
M.A. Nielsen and I. L. Chuang: Quantum Computation and Quantum Information
(Cambridge University Press)
N. D. Mermin: Quantum Computer Science (Cambridge University Press) K.F. Riley, M.P. Hobson & S.J. Bence, Mathematical Methods for Physics and Engineering (Cambridge University Press, 3^{rd} edition 2006).
Derek F. Lawden, Introduction to Tensor Calculus, Relativity and Cosmology (Dover 2002).
William D. D'Haeseleer, Jim Callen et al., Flux Coordinates and Magnetic Field Structure: A Guide to a Fundamental Tool of Plasma Theory (Springer-Verlag 1991).
Richard Fitzpatrick: Classical Electromagnetism lecture notes:
http://farside.ph.utexas.edu/teaching/em/lectures/node106.html