Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Society for Industrial and Applied Mathematics eBooks

Chapter 5: Discretization schemes

Alfio Borzı̀, Gabriele Ciaramella, M. Sprengel

Published Jul 6, 2017• 0 citations
Publisher

Abstract

In this chapter we discuss discretization schemes for the quantum models that are considered in this book. While we cannot provide an extensive discussion of the quite large number of approximation methods developed in computational mathematics and physics to solve quantum problems, we aim at illustrating those computational features that discretization schemes for controlled quantum systems should have in order to perform efficient simulation and compute accurate controls.

DOI
10.1137/1.9781611974843.ch5

Related papers

Chapter 7: Application to quantum control problems

Alfio Borzı̀, Gabriele Ciaramella, M. Sprengel

The purpose of this chapter is to revisit the quantum control problems and the solution methodologies discussed in the previous chapters presenting results of numerical experiments. In the following sections, we consider different model problems, discussed in Chapters 2, 3, and 4, and solution strategies based on the discretization schemes discussed in Chapter 5 and on the optimization methods illustrated in Chapter 6. We also provide some comparison of the strategies to provide information about which method is suitable in which context. Discussion on how to choose parameters characterizing the methods provide valuable insights for the practical implementation of quantum optimal control solvers.

Society for Industrial and Applied Mathematics eBooks•Jul 6, 2017•0 citations

Chapter 6: Hamiltonian simulation

Giacomo Nannicini

Hamiltonian simulation is the problem for which quantum computers were initially proposed by Richard Feynman [75], and it is a crucial problem for applications in physics: it corresponds to the simulation of the evolution of a quantum mechanical system over time. Although this book aims to be physics-free, it is useful to give at least the mathematical foundations of this problem, because Hamiltonian simulation is a central component of many quantum algorithms. We do not discuss the precise physical meaning of these concepts, and aim to provide just enough vocabulary to read the quantum computing literature without too much confusion.

Society for Industrial and Applied Mathematics eBooks•Dec 17, 2025•0 citations

Chapter 6: Numerical optimization methods

Alfio Borzı̀, Gabriele Ciaramella, M. Sprengel

This chapter is devoted to numerical optimization methods with a focus on solution schemes that are used in the framework of quantum control problems. We first discuss classical optimization schemes that require the computation of reduced gradients and usually provide first-order convergence to a local minimum. These methods are supported by a large set of theoretical results that provide guarantee of convergence to local minima. The second section is devoted to monotonic schemes that originate from the works of Vadim Fedorovitch Krotov. While these methods remain popular in the quantum control application communities, in other frameworks the methods of choice are those discussed in Section 6.1.

Society for Industrial and Applied Mathematics eBooks•Jul 6, 2017•0 citations

Quantum Mechanics Simulations

X. Zotos

This book and the accompanying software (PC Diskette) is a toolbox for performing a set of numerical simulations on simple quantum mechanical problems. It is one in a series of similar books (CUPS) covering different fields in physics from classical mechanics to nuclear and particle physics. Each chapter of the book is written by a different author and deals with a different physical problem. The choice of problems is standard, e.g. bound state and scattering problems in one and three dimensions, quantum mechanical time development; the exceptions are, a study of identical particles and electron states in a one dimensional lattice, an example from solid state physics. Although the chapters are self-contained and can be studied independently (at the expense of some minor duplication of material), care was taken with a uniform style of presentation and similar format in numerical simulations. As a result, it is easy to proceed in the study of the different problems having understood the technical possibilities of one simulation. Each chapter starts with a section presenting the basic theoretical concepts to be studied using the numerical simulation. It is followed by a section with a short description of the numerical technique used and an extensive series of suggestions for problems and quantities to experiment with, using the computer codes provided. A final section describes in detail how to run the computer programs. This book is not, and is not intended to be, a textbook for learning quantum mechanics nor a textbook of numerical analysis. Nevertheless, I find the theoretical presentation clear and straightforward enough for its purpose. The extensive and up to date list of references to books and research articles at the end of each chapter is also very helpful. Technically, I found the software easy to install and, after some initial testing, easy to run. The produced plots are clear and the use of units of atomic dimension gives a touch of reality to the simulations. In contrast to other similar books, this is not a `picture' book of quantum mechanics where results of numerical simulations are illustrated; here the reader is given the necessary means to play and discover for himself the properties of quantum mechanical systems. I believe that this book, used by teachers for demonstrations and students for independent study, provides a versatile tool for numerical experimentation in quantum mechanics.

European Journal of Physics•Mar 1, 1996•0 citations

Numerical simulations

Michel Le Bellac, Fabrice Mortessagne, G. G. Batrouni

As we have seen on several occasions, a great many of the physical phenomena of interest concern collective behaviour or strongly correlated particles. While it is sometimes possible to obtain useful insight using approximation methods, such as perturbation theory, reliable controlled approximations do not exist. In such situations, numerical simulations have become an indispensable tool. We shall not attempt in this chapter to discuss computer programming nor shall we discuss many of the powerful methods, both algorithmic and in data analysis, that have been developed since numerical simulations have become an important research tool. Instead, we shall assume that the reader already has enough familiarity with computer programming to apply the notions which we discuss here. Our goal is to explain enough of the basics of the Monte Carlo method (classical and quantum) to allow the reader to apply it readily to interesting equilibrium statistical physics problems of the type discussed in this book. The problems in this chapter have been tailored with this in mind. The programs needed are relatively simple and the physics illustrates many of the phenomena already seen: phase transitions with and without spontaneous symmetry breaking, critical exponents, diverging susceptibilities, scaling, vortices, superfluids etc. Consequently, some of the problems are somewhat long and may be better considered as mini-projects. Markov chains, convergence and detailed balance Even relatively innocent looking models are impossible to solve exactly. The exact solution of the two-dimensional Ising model is known but not in the presence of an external magnetic field. In three dimensions, even in the absence of a magnetic field, the exact solution is not known. In fact, relatively few models can be solved exactly; see Reference [14] for an excellent review of the subject.

Cambridge University Press eBooks•Apr 8, 2004•0 citations

Computational Methods for Simulating Quantum Computers

Hans A. De Raedt, Kristel F. L. Michielsen

This review gives a survey of numerical algorithms and software to simulate quantum computers. It covers the basic concepts of quantum computation and quantum algorithms and includes a few examples that illustrate the use of simulation software for ideal and physical models of quantum computers.

arXiv:quant-ph/0406210•Jun 27, 2004•36 citations
In this chapter we discuss discretization schemes for the quantum models that are considered in this book. While we cannot provide an extensive discussion of the quite large number of approximation methods developed in computational mathematics and physics to solve quantum problems, we aim at illustrating those computational features that discretization schemes for controlled quantum systems should have in order to perform efficient simulation and compute accurate controls.
The purpose of this chapter is to revisit the quantum control problems and the solution methodologies discussed in the previous chapters presenting results of numerical experiments. In the following sections, we consider different model problems, discussed in Chapters 2, 3, and 4, and solution strategies based on the discretization schemes discussed in Chapter 5 and on the optimization methods illustrated in Chapter 6. We also provide some comparison of the strategies to provide information about which method is suitable in which context. Discussion on how to choose parameters characterizing the methods provide valuable insights for the practical implementation of quantum optimal control solvers.
Hamiltonian simulation is the problem for which quantum computers were initially proposed by Richard Feynman [75], and it is a crucial problem for applications in physics: it corresponds to the simulation of the evolution of a quantum mechanical system over time. Although this book aims to be physics-free, it is useful to give at least the mathematical foundations of this problem, because Hamiltonian simulation is a central component of many quantum algorithms. We do not discuss the precise physical meaning of these concepts, and aim to provide just enough vocabulary to read the quantum computing literature without too much confusion.
This chapter is devoted to numerical optimization methods with a focus on solution schemes that are used in the framework of quantum control problems. We first discuss classical optimization schemes that require the computation of reduced gradients and usually provide first-order convergence to a local minimum. These methods are supported by a large set of theoretical results that provide guarantee of convergence to local minima. The second section is devoted to monotonic schemes that originate from the works of Vadim Fedorovitch Krotov. While these methods remain popular in the quantum control application communities, in other frameworks the methods of choice are those discussed in Section 6.1.
This book and the accompanying software (PC Diskette) is a toolbox for performing a set of numerical simulations on simple quantum mechanical problems. It is one in a series of similar books (CUPS) covering different fields in physics from classical mechanics to nuclear and particle physics. Each chapter of the book is written by a different author and deals with a different physical problem. The choice of problems is standard, e.g. bound state and scattering problems in one and three dimensions, quantum mechanical time development; the exceptions are, a study of identical particles and electron states in a one dimensional lattice, an example from solid state physics. Although the chapters are self-contained and can be studied independently (at the expense of some minor duplication of material), care was taken with a uniform style of presentation and similar format in numerical simulations. As a result, it is easy to proceed in the study of the different problems having understood the technical possibilities of one simulation. Each chapter starts with a section presenting the basic theoretical concepts to be studied using the numerical simulation. It is followed by a section with a short description of the numerical technique used and an extensive series of suggestions for problems and quantities to experiment with, using the computer codes provided. A final section describes in detail how to run the computer programs. This book is not, and is not intended to be, a textbook for learning quantum mechanics nor a textbook of numerical analysis. Nevertheless, I find the theoretical presentation clear and straightforward enough for its purpose. The extensive and up to date list of references to books and research articles at the end of each chapter is also very helpful. Technically, I found the software easy to install and, after some initial testing, easy to run. The produced plots are clear and the use of units of atomic dimension gives a touch of reality to the simulations. In contrast to other similar books, this is not a `picture' book of quantum mechanics where results of numerical simulations are illustrated; here the reader is given the necessary means to play and discover for himself the properties of quantum mechanical systems. I believe that this book, used by teachers for demonstrations and students for independent study, provides a versatile tool for numerical experimentation in quantum mechanics.
As we have seen on several occasions, a great many of the physical phenomena of interest concern collective behaviour or strongly correlated particles. While it is sometimes possible to obtain useful insight using approximation methods, such as perturbation theory, reliable controlled approximations do not exist. In such situations, numerical simulations have become an indispensable tool. We shall not attempt in this chapter to discuss computer programming nor shall we discuss many of the powerful methods, both algorithmic and in data analysis, that have been developed since numerical simulations have become an important research tool. Instead, we shall assume that the reader already has enough familiarity with computer programming to apply the notions which we discuss here. Our goal is to explain enough of the basics of the Monte Carlo method (classical and quantum) to allow the reader to apply it readily to interesting equilibrium statistical physics problems of the type discussed in this book. The problems in this chapter have been tailored with this in mind. The programs needed are relatively simple and the physics illustrates many of the phenomena already seen: phase transitions with and without spontaneous symmetry breaking, critical exponents, diverging susceptibilities, scaling, vortices, superfluids etc. Consequently, some of the problems are somewhat long and may be better considered as mini-projects. Markov chains, convergence and detailed balance Even relatively innocent looking models are impossible to solve exactly. The exact solution of the two-dimensional Ising model is known but not in the presence of an external magnetic field. In three dimensions, even in the absence of a magnetic field, the exact solution is not known. In fact, relatively few models can be solved exactly; see Reference [14] for an excellent review of the subject.