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Image Analysis Using Quantum Entropy Scale Space and Diffusion Concepts

Mark W. Coffey

Published Nov 1, 2009• 0 citations
OpenAlex

Abstract

Abstract : This report documents investigations of the connections between scale space and the linear and nonlinear diffusion of images, as well as investigations of concepts of generalized entropy and quantum entropy and their measures. The generalized entropies included in particular Renyi entropy with parameter q. The diffusion studies were performed with a combination of mathematical analysis and simulations. Analysis of quantum and semi-classical entropies of modeled physical systems was also performed. The feasibility of applying forms of generalized quantum search to scheduling and logistics problems was also investigated. As part of this effort, simulations of adaptive quantum search were performed. Furthermore, the capabilities and constraints of quantum lattice gas algorithms (QLGAs) were investigated, and a new QLGA for the telegraph equation with one spatial dimension was formulated, implemented, and simulated.

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This document aims to describe the methodology of quantum lattice gas algorithm based quantum physics simulations with details on implementation using graphics processing units. A variety of examples illustrating the general application of these methods are included. These examples are used to highlight topics such as scaling, potentials, momentums, multiple particle and dimensions, and various Hamiltonians in quantum simulations. Although intended to fullfill the doctor of philosophy degree requirement, this document is well suited as an introduction to this approach to modeling physics. In recent times parallelization has significantly improved the available computational power. Graphics processing units (GPUs) are a shining example of this, with thousands of compute cores and continually improved general computational capabilities, these processors can outshine their central processing unit (CPU) counterparts by orders of magnitude. Presented here are a number of examples of efficient computational methods for simulating physics using GPUs. Our quantum simulation method involves the quantum lattice gas model of quantum computation [59, 68, 64], however much of the parallel technology implementation is applicable to other numerical approaches. The quantum lattice gas model of computation is based on a unitary stream-collide quantum algorithm applied repeatedly over time upon an ordered array of qubits [53, 56, 68, 61, 63, 67]. The stream-collide quantum algorithm represents local quantum evolution of the system of quantum particles whose motion is constrained to reside on a spacetime lattice. Each particle in the system undergoes a quantum walk, interacting locally with other particles in the system that may reside on the same point. We will first mathematically describe the basic idea of quantum lattice gas algorithms. Next we will discuss GPUs and the basic features needed for implementation. We will then describe several examples of quantum lattice gas algorithm approach to illustrate the various applications and implementation challenges. Of particular interest is a look at modeling time-dependent many-body quantum mechanics using quantum lattice gas methods. For practical purposes we focus on the case of two fermions governed by the nonrelativistic Schrödinger equation. Numerical comparisons to analytic prediction show the high accuracy of these methods. Quantum lattice gas methods are chosen as a unitary algorithm for time dependent quantum simulation as they are efficient, accurate, and avoid the Fermi sign problem. Our findings from two-particle simulations under various conditions include the physics of the one and two particle systems diverging in their respective time-dependent behavior with increasing entanglement. We also confirm the multiple particle extension of the Ehrenfest theorem. Using the examples discussed throughout this document, we will then describe our specific computational implementation utilizing the computational power of GPUs through Nvidia’s compute unified device architecture (CUDA) language. Here we will discuss the open source software our group is developing based on these methods, along with some implementation verification and debugging concepts. Education is another application peripherally connected to this work and our software. The applications of this type of modeling as a learning tool is not fully explored yet, but the opportunities are promising.

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Abstract : This report documents investigations of the connections between scale space and the linear and nonlinear diffusion of images, as well as investigations of concepts of generalized entropy and quantum entropy and their measures. The generalized entropies included in particular Renyi entropy with parameter q. The diffusion studies were performed with a combination of mathematical analysis and simulations. Analysis of quantum and semi-classical entropies of modeled physical systems was also performed. The feasibility of applying forms of generalized quantum search to scheduling and logistics problems was also investigated. As part of this effort, simulations of adaptive quantum search were performed. Furthermore, the capabilities and constraints of quantum lattice gas algorithms (QLGAs) were investigated, and a new QLGA for the telegraph equation with one spatial dimension was formulated, implemented, and simulated.
The telegraph equation combines features of both the diffusion and wave equations and has many applications to heat propagation, transport in disordered media, and elsewhere. We describe a quantum lattice gas algorithm (QLGA) for this partial differential equation with one spatial dimension. This algorithm generalizes one previously known for the diffusion equation. We present an analysis of the algorithm and accompanying simulation results. The QLGA is suitable for simulation on combined classical-quantum computers.
Quantum entropy is a fundamental concept for quantum information recently developed in various directions. We will review the mathematical aspects of quantum entropy (entropies) and discuss some applications to quantum communication, statistical physics. All topics taken here are somehow related to the quantum entropy that the present authors have been studied. Many other fields recently developed in quantum information theory, such as quantum algorithm, quantum teleportation, quantum cryptography, etc., are totally discussed in the book (reference number 60).
The quantum lattice-gas automata (QLGA) have been studied to simulate the time-dependent Schrodinger equation. It is particularly well suited to implementation on a quantum computer; the many-body quantum system can be simulated very efficiently. On the other hand, its emulation on a classical computer would also have an advantage, because its unitarity might lead better behavior than standard finite-difference methods. In this study, we apply the QLGA method to the analysis of the one-particle quantum wave propagation in nanoscale devices. To this end, we develop a new algorithm to implement the open boundary condition into the QLGA method. In the QLGA simulation, the space is discretized and qubits are assigned to each node of the lattice. The each qubit contains the probability amplitude of whether the lattice node is occupied or not by the simulated particles. We have assigned the qubit array outside the simulation region to store the information about the particles which have flown out across the boundary. Then it becomes possible to simulate the open systems with the QLGA simulation.
This document aims to describe the methodology of quantum lattice gas algorithm based quantum physics simulations with details on implementation using graphics processing units. A variety of examples illustrating the general application of these methods are included. These examples are used to highlight topics such as scaling, potentials, momentums, multiple particle and dimensions, and various Hamiltonians in quantum simulations. Although intended to fullfill the doctor of philosophy degree requirement, this document is well suited as an introduction to this approach to modeling physics. In recent times parallelization has significantly improved the available computational power. Graphics processing units (GPUs) are a shining example of this, with thousands of compute cores and continually improved general computational capabilities, these processors can outshine their central processing unit (CPU) counterparts by orders of magnitude. Presented here are a number of examples of efficient computational methods for simulating physics using GPUs. Our quantum simulation method involves the quantum lattice gas model of quantum computation [59, 68, 64], however much of the parallel technology implementation is applicable to other numerical approaches. The quantum lattice gas model of computation is based on a unitary stream-collide quantum algorithm applied repeatedly over time upon an ordered array of qubits [53, 56, 68, 61, 63, 67]. The stream-collide quantum algorithm represents local quantum evolution of the system of quantum particles whose motion is constrained to reside on a spacetime lattice. Each particle in the system undergoes a quantum walk, interacting locally with other particles in the system that may reside on the same point. We will first mathematically describe the basic idea of quantum lattice gas algorithms. Next we will discuss GPUs and the basic features needed for implementation. We will then describe several examples of quantum lattice gas algorithm approach to illustrate the various applications and implementation challenges. Of particular interest is a look at modeling time-dependent many-body quantum mechanics using quantum lattice gas methods. For practical purposes we focus on the case of two fermions governed by the nonrelativistic Schrödinger equation. Numerical comparisons to analytic prediction show the high accuracy of these methods. Quantum lattice gas methods are chosen as a unitary algorithm for time dependent quantum simulation as they are efficient, accurate, and avoid the Fermi sign problem. Our findings from two-particle simulations under various conditions include the physics of the one and two particle systems diverging in their respective time-dependent behavior with increasing entanglement. We also confirm the multiple particle extension of the Ehrenfest theorem. Using the examples discussed throughout this document, we will then describe our specific computational implementation utilizing the computational power of GPUs through Nvidia’s compute unified device architecture (CUDA) language. Here we will discuss the open source software our group is developing based on these methods, along with some implementation verification and debugging concepts. Education is another application peripherally connected to this work and our software. The applications of this type of modeling as a learning tool is not fully explored yet, but the opportunities are promising.