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Representation of Cyclic Colour Spaces within Quantum Space

Michael NBernhard

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

Data of circular nature occurs frequently in several areas of science. In this paper we deal with data of this type, namely the HSI colour space. We show how one may embed this space into a quantum state space where distance measures exists that take the circular nature of the given data into account. This might hint that the quantum state space is useful not only for the problem at hand but may be used apart from quantum mechanics and quantum computations to solve classical problems as well.

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Data of circular nature occurs frequently in several areas of science. In this paper we deal with data of this type, namely the HSI colour space. We show how one may embed this space into a quantum state space where distance measures exists that take the circular nature of the given data into account. This might hint that the quantum state space is useful not only for the problem at hand but may be used apart from quantum mechanics and quantum computations to solve classical problems as well.
Quantum information theory is at the frontiers of physics, mathematics and information science, offering a variety of solutions that are impossible using classical theory. This book provides an introduction to the key concepts used in processing quantum information and reveals that quantum mechanics is a generalisation of classical probability theory. After a gentle introduction to the necessary mathematics the authors describe the geometry of quantum state spaces. Focusing on finite dimensional Hilbert spaces, they discuss the statistical distance measures and entropies used in quantum theory. The final part of the book is devoted to quantum entanglement - a non-intuitive phenomenon discovered by Schrödinger, which has become a key resource for quantum computation. This richly-illustrated book is useful to a broad audience of graduates and researchers interested in quantum information theory. Exercises follow each chapter, with hints and answers supplied.
Quantum computing is currently gaining significant attention, not only from the academic community but also from industry, due to its potential applications across several fields for addressing complex problems. For any practical problem which may be tackled using quantum computing, it is imperative to represent the data used onto a quantum computing system. Depending on the application, many different types of data and data structures occur, including regular numbers, higher-dimensional data structures, e.g., n-dimensional images, up to graphs. This report aims to provide an overview of existing methods for representing these data types on gate-based quantum computers.
In this paper, we investigate quantum algorithms for graph colouring problems, in particular for 2- and 3-colouring of graphs. Our main goal is to establish a set of quantum representations and operations suitable for the problem at hand. We propose unitary- as well as measurement-based quantum computations, also taking inspiration from answer set programming, a form of declarative programming close to traditional logic programming. The approach used is one in which we first generate arbitrary solutions to the problem, then constraining these according to the problem's input. Though we do not achieve fundamental speed-ups, our algorithms show how quantum concepts can be used for programming and moreover exhibit structural differences. For example, we compute all possible colourings at the same time. We compare our algorithms with classical ones, highlighting how the same type of difficulties give rise to NP-complete behaviour, and propose possible improvements.
We discuss some basic tools for an analysis of one-dimensional quantum systems defined on a cyclic coordinate space. The basic features of the generalized coherent states, the complexifier coherent states, are reviewed. These states are then used to define the corresponding (quasi)densities in phase space. The properties of these generalized Husimi distributions are discussed, in particular their zeros. Furthermore, the use of the complexifier coherent states for a semiclassical analysis is demonstrated by deriving a semiclassical coherent state propagator in phase space.