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Lyle Noakes

Publications and source records attributed to Lyle Noakes.

26 records · Page 2Linked to original sources
arXiv2017

Generating 3 qubit quantum circuits with neural networks

A new method for compiling quantum algorithms is proposed and tested for a three qubit system. The proposed method is to decompose a a unitary matrix U, into a product of simpler U j via a neural network. These U j can then be decomposed into product of known quantum gates. Key to the effectiveness of this approach is the restriction of the set of training data generated to paths which approximate minimal normal subRiemannian geodesics, as this removes unnecessary redundancy and ensures the products are unique. The two neural networks are shown to work effectively, each individually returning low loss values on validation data after relatively short training periods. The two networks are able to return coefficients that are sufficiently close to the true coefficient values to validate this method as an approach for generating quantum circuits. There is scope for more work in scaling this approach to larger quantum systems.

quant-ph↗
arXiv2016

Low Energy Clamped Planar Elastica

A method is given for estimating clamped plane elastica. Arguments are made, and evidence is provided by way of illustrative examples, suggesting that the new method is quicker and more robust than standard discretisation, and more likely to give elastica of low energy. Just as for standard discretisation, no use is made of the known solutions for elastica in terms of elliptic functions.

math.GM↗
arXiv2014

Travelling times in scattering by obstacles

The paper deals with some problems related to recovering information about an obstacle in an Euclidean space from certain measurements of lengths of generalized geodesics in the exterior of the obstacle. The main result is that if two obstacles satisfy some generic regularity conditions and have (almost) the same traveling times, then the generalized geodesic flows in their exteriors are conjugate on the non-trapping part of their phase spaces with a time preserving conjugacy. In the case of a union of two strictly convex domains in the plane, a constructive algorithm is described to recover the obstacle from traveling times.

math-ph↗
arXiv2013

Relative Geodesics in the Special Euclidean Group

We propose a notion of distance between two parametrized planar curves, called their discrepancy, and defined intuitively as the minimal amount of deformation needed to deform the source curve into the target curve. A precise definition of discrepancy is given as follows. A curve of transformations in the special Euclidean group SE(2) is said to be admissible if it maps the source curve to the target curve under the point-wise action of SE(2) on the plane. After endowing the group SE(2) with a left-invariant metric, we define a relative geodesic in SE(2) to be a critical point of the energy functional associated to the metric, over all admissible curves. The discrepancy is then defined as the value of the energy of the minimizing relative geodesic. In the first part of the paper, we derive a scalar ODE which is a necessary condition for a curve in SE(2) to be a relative geodesic, and we discuss some of the properties of the discrepancy. In the second part of the paper, we consider discrete curves, and by means of a variational principle, we derive a system of discrete equations for the relative geodesics. We finish with several examples.

math.DG↗
arXiv2011

Conditional Extremals

Imagine that measurements are made at times $t_0$ and $t_1$ of the trajectory of a physical system whose governing laws are given approximately by a class ${\cal A}$ of so-called {\em prior vector fields}. Because the physical laws are not known precisely, it might be that the measurements are not realised by the integral curve of any prior field. We want to estimate the behaviour of the physical system between times $t_0$ and $t_1$. An integral curve of an arbitrary vector field $X$ is said to be {\em feasible} when it interpolates the measurements. When $X$ is critical for $L^2$ distance to ${\cal A}$, the feasible curve is called a {\em conditional extremum}. When the distance to ${\cal A}$ is actually minimal, the conditional extremum is a best estimate for the intermediate behaviour of the system. The present paper does some of basic groundwork for investigating mathematical properties of conditional extrema, focusing on cases where ${\cal A}$ comprises a single prior field $A$. When $A={\bf 0}$ a conditional extremal is a geodesic arc, but this special case is not very representative. In general, $A$ enters into the Euler-Lagrange equation for conditional extrema, and more can be said when $A$ is conservative or has special symmetry. We characterise conservative priors on simply-connected Riemannian manifolds in terms of their conditional extrema: when time is reversed, a constant is added to the $L^2$ distance. For some symmetric priors on space forms we obtain conditional extrema in terms of the Weierstrass elliptic function. For left-invariant priors on bi-invariant Lie groups, conditional extrema are shown to be right translations of pointwise-products of 1-parameter subgroups.

math.DG↗
arXiv2011

Minimum $L^\infty$ Accelerations in Riemannian Manifolds

Riemannian cubics are critical points for the $L^2$ norm of acceleration of curves in Riemannian manifolds $M$. In the present paper the $L^\infty$ norm replaces the $L^2$ norm, and a less direct argument is used to derive necessary conditions analogous to those for Riemannian cubics. The necessary conditions are examined when $M$ is a sphere or a bi-invariant Lie group.

math.DG↗
arXiv2011

Nearly Geodesic Riemannian Cubics in SO(3)

{\em Riemannian cubics} are curves in a manifold $M$ that satisfy a variational condition appropriate for interpolation problems. When $M$ is the rotation group SO(3), Riemannian cubics are track-summands of {\em Riemannian cubic splines}, used for motion planning of rigid bodies. Partial integrability results are known for Riemannian cubics, and the asymptotics of Riemannian cubics in SO(3) are reasonably well understood. The mathematical properties and medium-term behaviour of Riemannian cubics in SO(3) are known to be be extremely rich, but there are numerical methods for calculating Riemannian cubic splines in practice. What is missing is an understanding of the short-term behaviour of Riemannian cubics, and it is this that is important for applications. The present paper fills this gap by deriving approximations to nearly geodesic Riemannian cubics in terms of elementary functions. The high quality of these approximations depends on mathematical results that are specific to Riemannian cubics.

math.DG↗