# Annotated Source List

November 27, 2015

**Ando, Y., & Itoh, T. (1987). Calculation of transmission tunneling current across arbitrary potential barriers. Journal of Applied Physics, 61(4).**

This paper presents a straightforward method to calculate transmission probability and current across arbitrary potential barriers using an approximation with many rectangles (multistep approximation). This is applicable to various barriers and they are analyzed to demonstrate the use of this method. In the present calculation, instead of dealing with continuous variations in the potential energy, it splits the potential barrier into segments.

This paper is especially useful for my project because it provides a method to calculate the transmission coefficient of an arbitrary potential. The solution for even a triangular barrier includes Airy functions but they are unsuitable for designing quantum-well structures due to the complex treatment. Multistep functions can be easily implemented to solve the same problem.

**Calderon, F. (n.d.). [Personal interview by the author].**

My mentor’s graduate student is Mr. Calderon. He works in the field of Quantum Information in the Department of Physics at the University of Maryland, Baltimore County. Mr. Calderon is working to obtain his Ph.D. I can contact him for assistance when I am learning or working on Dr. Kestner’s problems.

**Griffiths, D. J. (1995). Introduction to quantum mechanics. Upper Saddle River, NJ: Prentice Hall.**

The textbook is separated into two parts. Part 1 describes the theory behind quantum mechanics. Part 2 discusses applications of the theory. The first chapter explains the difference between quantum mechanics and classical mechanics. It describes what a wave function is. The second chapter introduces basic models for introductory quantum mechanics. Chapter 3 covers linear algebra in which quantum mechanics is formulated. Chapters 4 and 5 expand upon the basic models described in chapter 2. Part 2 describes many different approximation techniques used to find the solution, when solving the equation analytically is impossible.

This introductory textbook to quantum mechanics is especially useful for learning the basics of quantum physics. This text provides the background information necessary for me to work on the research project. I find it very useful being able to consult it as reference. Griffiths explained the concepts clearly and the textbook is heralded as a classic in the field.

**Kaye, P. R., Laflamme, R., & Mosca, M. (2007). An introduction to quantum computing. New York, NY: Oxford University Press.**

The textbook begins with an introduction to computers, quantum physics, and the formalism in linear algebra. It expands upon the idea of qubits and a quantum model of computation. Then it discusses introductory quantum algorithms, complexity theory, and quantum error correction.

This textbook provides an introduction to quantum computing which is the field my mentor works in. It may be useful for me to study as I work more on the research project. Kaye starts the book from the basics so it would be easy for the reader to understand the ideas to make progress. The mathematics is build up so that the reasoning is logical and coherent.

**Kestner, J. (n.d.). [Personal interview by the author].**

My mentor is Dr. Jason Kestner. He works in the field of Quantum Information in the Department of Physics at the University of Maryland, Baltimore County. He is interested in developing physical devices based upon quantum mechanics, namely, the quantum computer. He received his B.S. in Physics at Michigan Technological University in 2004 and Ph.D. in Physics at the University of Michigan in 2009. He was previously a postdoctoral researcher in the Condensed Matter Theory Center of the University of Maryland, College Park.

**Knill, E., Laflamme, R., & Milburn, G. J. (2001). A scheme for efficient quantum computation with linear optics. Nature, 409.**

This journal starts with an introduction to quantum computation and describes the applications of quantum computing to solving problems. The main goal is to implement the quantum elements in a physical system to demonstrate that computation is possible. Knill et. al show that quantum computation is possible using beam splitters, phase shifters, single photon sources, and photo-detectors. This method is accessible to investigation with current technology. One can use a resonant tunneling scheme to perform a joint measurement on the spins via conductance. The Knill-Laflamme-Milburn proposal can be used to entangle spatially separated spins. This is important because the spins do not need to interact with each other at all.

This journal is indirectly important to my research project because the proposal is based off of this work. Previous attempts to entangle spins required coupling charges to a single quantum point contact, requiring the dots to be very close to each other. The combination of resonant tunneling and measurement-based quantum information processing are now placed in the context of each other.

**Lay, D. C. (2012). Linear Algebra and Its Applications (4th ed.).**

** Addison-Wesley.**

This textbook introduces the mathematics of linear algebra. Chapter 1 discusses the importance of solving linear equations in real-life situations and describes the applications of linear algebra in analyzing these systems of equations. Chapter 2 covers matrix algebra in depth and the operations and capabilities that computing with matrices allow. Determinants are introduced in chapter 3 with their significance and applications thoroughly described. Chapter 4 is more relevant to my work because it clarifies the concepts of vector spaces applicable to the notation I have been seeing. It describes the concept of a vector which may initially not seem related to linear algebra. Chapter 5 is on one of the most important concepts in linear algebra applicable to physics, eigenvalues and eigenvectors. The main concept is to find vectors which are only changed by a constant scalar under a matrix transformation. Chapter 6 covers orthogonality in vectors which is commonly used in quantum computation, to re-express a vector in terms of a different set of basis vectors. Chapter 7 describes symmetric matrices and the spectral decomposition which are included in the Nielsen and Chuang introduction. The textbook concludes in chapter 8 by describing the geometry of vector spaces, applying linear algebra to geometric figures. This section is more applicable to cover the mathematical details of linear algebra. The appendices review complex numbers used in the linear algebra formulation.

In my study of the Nielsen and Chuang quantum computation textbook, a broad understanding of linear algebra is required to understand the formulation behind qubits. My mentor recommended me to use this book as a reference in my studies so I would be able to use this textbook to learn linear algebra techniques for quantum computation. This supplementary text is easy to follow and explains the concepts in depth. This undergraduate textbook includes multiple examples to allow for better understanding of abstractions. Chapters 5-7 are most relevant to my project. This resource is very useful for building my mathematical foundation, in addition to differential equations, for learning quantum computation.

**Nielsen, M. A., & Chuang, I. L. (2010). Quantum computation and quantum information (Tenth Anniversary ed.). Cambridge University Press.**

This textbook serves as a guide to understanding quantum computation and quantum information processing. It describes what a quantum computer is and how it can be used to solve problems faster than classical computers. Chapter 1 gives a global perspective on quantum computing. The concept of qubits and quantum gates are previewed and presented similarly to classical computation. There are remarks on the possibilities and applications of quantum computation. Chapter 2 outlines the more rigorous introduction to the necessary linear algebra and quantum mechanics background needed to work through the text. Chapter 3 describes fundamentals of classical computer science so the knowledge may be applied to quantum computer science. Quantum circuits are covered in Chapter 4 with operations on single qubits and multiple qubits. The method for simulation of quantum systems is outlined in this section. Chapter 5 described one application where quantum computing is faster than classical computing: the quantum Fourier transform. This offers an exponential speed-up to factoring large numbers and finding discrete logarithms. Chapter 6 describes another application in quantum search algorithms. Chapter 7 is significant in that it describes physical realizations for quantum computers. Chapters 8 and 9 discuss quantum noise and operations which are important in real-life situations. Chapter 10 is most related to my work because it thoroughly describes quantum error-correction. Joint measurement has important applications to quantum error-correction. Chapters 11 and 12 are more general, focusing on abstract quantum information theory.

My mentor recommended me this textbook and assigned me readings in sections 1.2-1.4, 2.1-2.2, and 4.1-4.6. This book is especially relevant to the project I’m working on in regards to understanding the joint measurement process. With a broad and deep understanding of quantum computation, I will be able to explore further possibilities and make significant process on this project. It is a well-known textbook for research in physics, computer science, mathematics, and electrical engineering. I find this source to be very useful for learning about quantum computation.

**Rozman, M. G., Reineker, P., & Tehver, R. (1994). One-dimensional scattering: Recurrence relations and differential equations for transmission and reflection amplitudes. Physical Review A, 49(5).**

This paper develops a recurrence method for analytical and numerical evaluation of tunneling, transmission, and reflection amplitudes. It describes recurrence formulas which express the scattering amplitudes for the arbitrary segmented potential into subparts. This can be applied to any potential, giving differential equations for transmission, and reflection amplitudes.

My mentor sent me this article to better understand one-dimensional scattering as well as to verify my algorithm for the double-barrier potential. It gives a thorough account of one-dimensional scattering which has important applications to my project. I can check the development of my algorithm with the calculated results described in the paper.