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Quantum Mechanics, Quantum Computation, and the Density Operator in SymPy

Addison Cugini

Published Jan 1, 2011• 1 citation
OpenAlex

Abstract

Because aspects of quantum mechanics are both difficult to understand and difficult algebraically, there is a need for software which symbolically simulates quantum me-chanical phenomena. To accomplish this goal, code has been added to an open-source computer algebra system, called SymPy, which models the abstraction of Dirac nota-tion and the density operator. Additionally, a quantum computer simulation has been built using this abstraction. This paper shall discuss the code that has been added as well as any relevant physics important to understanding the code. In particular, we shall focus on the density operator of statistical quantum mechanics using SymPy's density matrices as a guide. 1

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Because aspects of quantum mechanics are both difficult to understand and difficult algebraically, there is a need for software which symbolically simulates quantum me-chanical phenomena. To accomplish this goal, code has been added to an open-source computer algebra system, called SymPy, which models the abstraction of Dirac nota-tion and the density operator. Additionally, a quantum computer simulation has been built using this abstraction. This paper shall discuss the code that has been added as well as any relevant physics important to understanding the code. In particular, we shall focus on the density operator of statistical quantum mechanics using SymPy's density matrices as a guide. 1
Quantum Mechanics can be an abstract and complex subject. Students often complain of confusion, struggle, and frustration as they try to master the topic. The goal of this book is to reduce the complexity and clarify the abstractions with concrete visual examples driven by simple python programs. It is assumed that the reader is concurrently taking a course in quantum mechanics, or self-studying quantum mechanics, but is looking for supplementary material to help with understanding and visualizing how quantum mechanics works. The focus of this book is writing python programs to visualize the underlying behavior of the mathematical theory. The background needed to understand quantum mechanics is differential equations, linear algebra and modern physics. We need a strong foundation in differential equations and linear algebra because the behavior of quantum systems is governed by equations that are written in terms of these concepts. Modern physics includes concepts such as special relativity and quantum phenomena like the photoelectric effect and energy quantization that the theory of quantum mechanics seeks to explain. This book is also not an introduction to the python programming language, or to numpy, or even to VPython. However its programming examples start simply and grow more complex as the chapters progress, so deep expertise in any of these is not a pre-requisite. Key features: · Provides an accessible and practical guide to the abstractions in quantum mechanics with concrete visual examples driven by simple python programs. · Contains few derivations, equations, and proofs. · For complete beginners of python programming, appendix B serves as a very brief introduction to the main concepts needed to understand the code in this book.
Q UANTUM MECHANICS can be an abstract and complex subject. Students often complain of confusion, struggle, and frustration as they try to master the topic. The goal of this book is to reduce the complexity and clarify the abstractions with concrete visual examples driven by simple python programs. It is assumed that the reader is concurrently taking a course in quantum mechanics, or self-studying quantum mechanics, but is looking for supplementary material to help with understanding and visualizing how quantum mechanics works. There are few derivations, equations, or proofs. Those are left to the more formal textbooks and lectures. The focus of this book is writing python programs to visualize the underlying behavior of the mathematical theory. The background needed to understand quantum mechanics is differential equations, linear algebra and modern physics. We need a strong foundation in differential equations and linear algebra because the behavior of quantum systems is governed by equations that are written in terms of these concepts. Modern physics includes concepts such as special relativity and quantum phenomena like the photoelectric effect and energy quantization that the theory of quantum mechanics seeks to explain. This book is also not an introduction to the python programming language, or to numpy, or even to VPython. Having said that, the programming examples will start simply and grow more complex as the story line progresses, so deep expertise in any of these is not a pre-requisite. If you're completely new to python programming, appendix B has been provided as a very brief introduction to the main concepts needed to understand the code in this book.
A common problem when learning Quantum Mechanics is the complexity in the mathematical and physical concepts, which leads to difficulty in solving and understanding problems. Using programming languages like Python have become more and more prevalent in solving challenging physical systems. An open-source computer algebra system, SymPy, has been developed using Python to help solve these difficult systems. I have added code to the SymPy library for two different systems, a One-Dimensional Quantum Harmonic Oscillator and a Quantum Mapping Gate used in Quantum Computing.
After the emergence of quantum mechanics and realising its need for an accurate understanding of physical systems, numerical methods were being used to undergo quantum mechanical treatment. With increasing system correlations and size, numerical methods fell rather inefficient, and there was a need to simulate quantum mechanical phenomena on actual quantum computing hardware. Now, with noisy quantum computing machines that have been built and made available to use, realising quantum simulations are edging towards a practical reality. In this paper, we talk about the progress that has been made in the field of quantum simulations by actual quantum computing hardware and talk about some very fascinating fields where it has expanded its branches, too. Not only that, but we also review different software tool-sets available to date, which are to lay the foundation for realising quantum simulations in a much more comprehensive manner.
In the field of quantum information science, one can design a series of quantum logic operations known as a circuit. Circuits are the basis for quantum computations in quantum computing. As circuits will most likely be designed from a logical standpoint, there could exist mathematical redundancies which will lead to a larger circuit than necessary. These redundancies are computationally expensive, and there is a need for them to be found and eliminated to simplify the circuit. We present our research on finding the rules for simplifying circuits and its implementation in SymPy.