Advanced Theoretical Techniques & Introduction to Quantum Computing - PHY00044H

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  • Department: Physics
  • Module co-ordinator: Prof. Irene D'Amico
  • Credit value: 20 credits
  • Credit level: H
  • Academic year of delivery: 2018-19
    • See module specification for other years: 2019-20

Related modules

Co-requisite modules

  • None

Prohibited combinations

  • None

Module will run

Occurrence Teaching cycle
A Autumn Term 2018-19 to Spring Term 2018-19

Module aims

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.

Module learning outcomes

  • discuss the fundamentals of quantum computation: concept of quantum bit (qubit); single and two qubit gates; role of superposition principle (quantum parallelism); concept of entanglement; concept of density matrix and its properties; differences between pure and mixed states
  • understand and been able to use circuit representation of quantum gates
  • understand and describe some of the quantum algorithms and the basics of quantum-error correction
  • understand teleportation and describe the simplest teleportation protocol
  • describe the requirements for physical systems to be used as quantum computers;
  • understand the main physical limitations to quantum computation (decoherence and scalability); understand how decoherence influences density matrices
  • describe specific proposals on quantum computers
  • understand and describe some experimental results related to specific proposals
  • describe basic ideas behind ‘one-way’ quantum computing
  • calculate the Fourier transform (and inverse transform) of a given function
  • calculate the convolution of two functions
  • calculate the Laplace transform of a given function
  • use Fourier and Laplace transforms to solve partial differential equations
  • use the calculus of variations to solve problems in mechanics and find extremal solutions in other fields such as geometry (shortest path, geodesics) and statistics (maximum entropy).
  • use Lagrange multipliers in the calculus of variations to solve problems with constraints
  • understand the correspondence between symmetries and conservation laws in physical theories
  • identify symmetries in a functional, and find the corresponding conservation law
  • understand and describe how the calculus of variations extends to field theories
  • Efficiently manipulate multidimensional mathematical objects
  • Apply anisotropic physics models and interpret their physical consequences (through a variety of examples)
  • Identify and apply spatial transformations, including three-dimensional rotations
  • Distinguish between tensorial and non-tensorial objects
  • Demonstrate invariance of various quantities in both Euclidean and Minkowski spaces
  • Manipulate tensorial quantities (both covariant and contravariant) in non-orthogonal coordinate systems
  • Solve a range of unseen problems based on the syllabus given below

Module content

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.

Assessment

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

Special assessment rules

None

Reassessment

Task Length % of module mark
University - closed examination
Advanced Theoretical Techniques & Introduction to Quantum Computing
3 hours 86

Module feedback

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.

Indicative reading

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, 3rd 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



The information on this page is indicative of the module that is currently on offer. The University is constantly exploring ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary by the University. Where appropriate, the University will notify and consult with affected students in advance about any changes that are required in line with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.