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Title page!
Describes the path the
talk will take, to be summarized at the end of each significant chunk of the presentation.
Explains the basic
novelty in coming up with the ideas for this work – principally the use of a quantum rather than a classical physical model for the
optimization problem.
Explains pictorially the
principal property exhibited in the Schrödinger equation which is not present in the diffusion equation!
Demonstration of a
Crank-Nicholson numerical time-integration of the Schrödinger equation for a 6th-order
polynomial potential.
Will be surrounded by some
talk about why it is so unfeasible to use quantum dynamics
for an optimization algorithm.
Update on where we are in the conceptual
path of the presentation.
Description of particle
representation and dynamics in Quantum Mechanics.
Description of solutions to the
Schrödinger equation, including the specific form of the solutions for a
harmonic oscillator potential.
The key theorem underpinning the
algorithm – large mass limit localization of the ground state. Specific description of the full
perturbation theory approach to approximating solutions for the eigenfunctions
of a system with a general potential.
Description of use of the harmonic
oscillator approximation, and that its use in perturbation theory allows
approximate solution for any polynomial potential.
Update on where we are in the conceptual path of the presentation.
Performing perturbation
theory with an harmonic oscillator approximation as described to approximate the ground state of a 1-d quartic polynomial
potential – 3 basisi functions localized to the
left-hand well of the potential.
Approximate ground state
using 7 basis functions localised to the left hand potential well.
The same using 30 HO
basis functions.
This time using 30 HO
basis functions localized to the right hand well, as a sanity-check.
Visual demonstration of
successful solutions to a 2-dimensional quartic polynomial problem.
2d-groundstate
approximation using 500 basis functions.
Solutions ofr the 2d
potential using all 3 local minima as centres of the HO basis set.
Update on where we are in the conceptual path of the presentation.
Description of the use
of diagonalization/factorization of the harmonic oscillator basis set, and the ladder operator tricks we use to make this tractable
in high-dimensional problems.
An explanation of how we use the fact
that solutions of interest to binary problems are on the corners of the
hypercube, to select the HO basis set.
A description of the
prototype problem used to understand the spatial-spanning property of the harmonic oscillator basis set.
Mathematical follow
through of spatial span of HO basis functions, giving the present best upper bound on the complexity of the algorithm for general
problems.
An empirical view of the sparseness of
use of the basis functions in an example of factoring the bi-prime – 35.
Update on where we are in the conceptual path of the presentation.
A brief explanation that any member of
the set of computationally hard problems can be encoded as a fourth-order
polynomial potential function, and how polynomial potential functions can be
dealt with efficiently for the purposes of computing an approximate ground
state for their corresponding QM system.
Brief description of the similarities
and differences between the Schrödinger and Diffusion equations, in the
context of implementing a Tunneling Salesman-like algorithm based upon the
Diffusion equation.
Classical vs. Quantum Contd.
Update on where we are in the conceptual path of the presentation.
An explicit description of encoding of
the bi-prime factoring problem, and a statement of successful results.
Update on where we are in the conceptual path of the presentation.
My conclusions!
My acknowledgements.