nach oben

2025 | Buch

A First Introduction to Quantum Computing and Information

verfasst von: Bernard Zygelman

Verlag: Springer International Publishing

Buchreihe : Undergraduate Topics in Computer Science

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Inhaltsverzeichnis

Über dieses Buch

This textbook addresses and introduces new developments in the field of quantum information and computing (QIC) for a primary audience of undergraduate students.

Developments over the past few decades have spurred the need for QIC courseware at major research institutions. To address this, the present 2nd edition of a highly accessible textbook/reference broadens the exposure of QIC science for the undergraduate market. The subject matter is introduced in such a way so that it is accessible to students with only a first-year calculus background. Greater accessibility allows a broader range of academic offerings.

Topics and features:

Introduces beginning undergraduate students to quantum theory and developments in QIC, without exposure to upper-level physics and mathematics Contains a new chapter on Adiabatic Quantum Computing Integrates Mathematica-based software examples and projects, which offers a “hands-on" experience and facilitates navigation of difficult abstract concepts Offers helpful links to additional exercises, problems, and solution manuals Facilitates a more holistic approach to the teaching of difficult concepts, incorporating symbolic manipulation software Provides new material on Quantum Error Correction Allows a broad-range of course offerings spanning physics, engineering, math and computer science

This unique introductory textbook can serve courses offered in university physics, engineering, math, and definitely computer science departments. Use of Mathematica software allows a fast learning curve for students who have limited experience with scientific programming.

Inhaltsverzeichnis

Frontmatter

Chapter 1. A Quantum Mechanic’s Toolbox

Abstract

The concepts of bit and qubit, the fundamental units of information in classical and quantum computing respectively, are introduced. We discuss features of the binary number system, linear vector, and Hilbert spaces. We learn how to manipulate qubits and introduce Dirac’s bra-ket formalism to facilitate operations in Hilbert space. Scalar, direct, and outer products of bra-kets in multi-qubit systems are introduced. We define Hermitian and unitary operators in a \(2^n\)-dimensional Hilbert space and use the bra-ket formalism to construct them. I summarize the foundational postulates of quantum mechanics, as espoused by the Copenhagen interpretation, for a finite set of qubits.

Bernard Zygelman

Chapter 2. Apples and Oranges: Matrix Representations

Abstract

I discuss and elaborate on the isomorphism between kets that span the Hilbert space of n-qubits with column matrices of dimension \(2^{n}\). The collection of all row matrices is shown to constitute the corresponding dual space. We illustrate how outer products, or operators, are represented by \(n \times n \) square matrices. The various matrix operations that provide the inner, outer, direct or Kronecker, products for the corresponding Hilbert space are introduced and discussed. The concepts of spin and the Bloch sphere are introduced. A qubit interpretation of spin, and the polarization properties of light is discussed.

Bernard Zygelman

Chapter 3. Circuit Model of Computation

Abstract

We provide a brief review of Boolean logic, the circuit model of computation, and I show how to assemble logic gates from their Boolean components. I present examples of classical circuits, including the half, full and ripple adder, and discuss reversible and irreversible gates. Quantum logic gates, including the Pauli, Hadamard and controlled-not gates are introduced, and we learn how to construct quantum circuits from them. The notions of quantum parallelism and interference enable Deutsch’s algorithm, the first proof-of-principle for quantum advantage. We dissect the quantum circuit for the Deutsch-Josza algorithm and demonstrate its ability to perform massively parallel computations; a capability inaccessible to machines based on the bit paradigm. We introduce and discuss the unitary time development of quantum states.

Bernard Zygelman

Chapter 4. Quantum Killer Apps: Quantum Fourier Transform and Search Algorithms

Abstract

We present a brief overview of the Fourier series, the Fourier and discrete Fourier transforms and their applications. We discuss a quantum algorithm that encodes the Fourier transform of the mapping \( f:\{0,1\}^{n} \rightarrow \{0,1\}\) in an n-qubit register. It’s shown how the quantum Fourier transform (QFT) gate is constructed from single-qubit phase and two-qubit control gates. Due to the collapse postulate, the quantum Fourier transform for f is not available in a register query, but it does allow efficient period estimation. We illustrate how the QFT is exploited in the Shor algorithm for factoring large numbers. On the average, search for an item in an unordered list of size N requires N/2 queries. We show how the Grover quantum algorithm improves on this figure of merit as it requires resources that scale as \(\sqrt{N}\).

Bernard Zygelman

Chapter 5. Quantum Mechanics According to Martians: Density Matrix Theory

Abstract

We discuss the density matrix approach introduced by John von Neumann to quantum mechanics. We illustrate how it is applied in the calculation of expectation values and Born probabilities. The postulates of quantum mechanics, according to the Copenhagen interpretation, are summarized from the vantage point of the density operator framework. We learn how density operators are used to discern coherent, or pure, state ensembles from statistical mixtures of pure states. It is shown that, for entangled states, the traced density operator to a lower dimensional Hilbert space, results in reduced density operators that describe a mixed state. The Schmidt decomposition theorem and the Schmidt number, which measures the degree of entanglement of quantum states, is introduced and discussed. We define and illustrate the concept of von Neumann entropy.

Bernard Zygelman

Chapter 6. No-Cloning Theorem, Quantum Teleportation and Spooky Correlations

Abstract

The no-cloning theorem states that an arbitrary quantum state cannot be copied from one qubit and duplicated on another qubit. We offer a proof of this theorem and illustrate how quantum states can be teleported between two qubits. The Bell and Clauser-Horne-Shimony-Holt inequalities are introduced and shown to be demonstrable features of entangled quantum systems. We discuss how the private key distribution problem is dealt with using quantum key distribution (QKD). The BB84 and Ekert protocols are examples of the latter, and we review and illustrate their implementation. We show how entangled states enable dense coding and offer a brief synopsis of Greenberger-Horne-Zeilinger (GHZ) states and their application.

Bernard Zygelman

Chapter 7. Quantum Hardware I: Ion Trap Qubits

Abstract

The DiVincenzo criteria, a list of necessities for the construction of a quantum computer is summarized herein. In reviewing the physics underpinning the trapped ion qubit paradigm, I introduce a rotor model for atoms/ions and demonstrate how single qubit gates, such as the phase and Hadamard gates, are realized in this framework. We use the latter to illustrate how ions respond to laser radiation, the mechanism by which ion qubits are addressed. We show how the Cirac-Zoller mechanism, and its generalization, enables the realization of two-qubit control gates. I provide a survey of trapped ion qubit systems, including optical and hyperfine qubit systems.

Bernard Zygelman

Chapter 8. Quantum Hardware II: cQED and cirQED

Abstract

We introduce the vacuum Maxwell equations and use them to describe electromagnetic fields trapped in a cavity. Boundary conditions for the cavity are shown to lead to standing wave solutions which we quantize to construct cavity QED, a quantum theory for those fields. We insert a rotor into the cavity and are led to a quantum description of a rotor(atom) qubit coupled to a quantized electromagnetic field. We derive the Jaynes-Cummings Hamiltonian and find its approximate eigenvalues and eigenvectors in the strong atom-radiation coupling regime. We show how artificial atoms, composed of superconducting Josephson junctions, interact with microwave line-resonator photons, thus allowing a circuit analog of cavity QED. We discuss how the latter is described with electrical circuit terminology.

Bernard Zygelman

Chapter 9. Computare Errare Est: Quantum Error Correction

Abstract

We show how to diagnose and rehabilitate bit-flip errors in a classical error correcting model. We compare logical versus physical bits, define codewords, and introduce a classical error-correcting model. We illustrate elements of the latter, including logical versus physical bits, codewords and the notion of an error correcting code. Quantum codes must consider the no-cloning theorem, the collapse hypothesis, and the possibility of continuous errors. We present encoding, syndrome measurement, and recovery circuits for single qubit bit-flip and phase shift errors. We review the Shor code, introduce the stabilizer formalism, and illustrate stabilizers role in its implementation. We demonstrate the use of the stabilizer formalism in the analysis of quantum error-detection in the Laflamme and Steane codes and the development of surface codes. We discuss the threshold theorem and its role in allowing for fault-tolerant quantum computing.

Bernard Zygelman

Chapter 10. Adiabatic Quantum Computing

Abstract

This chapter introduces Adiabatic Quantum Computing (AQC) and highlights its potential as an alternative to the circuit model. We introduce, via numerical demonstration, the quantum adiabatic theorem, which asserts that a system will remain in its ground state provided the Hamiltonian changes slowly enough. This principle forms the basis of AQC’s approach, making it capable of addressing complex problems like determining the ground states of quantum many-body systems, or the extremum of a cost function. The chapter covers the technique of embedding a time-independent Hamiltonian in a time-dependent one, facilitating a transition from a simple, solvable system to a more complex target problem. By slowly evolving the system, the initial Hamiltonian’s ground state becomes the target Hamiltonian’s ground state. Additionally, the text explores its application in areas traditionally linked to the circuit model, like Grover’s algorithm. Theoretical concepts are demonstrated with practical examples. Lastly, the chapter examines geometric and topological phases generated by adiabatic evolution and explores their potential as a resource.

Bernard Zygelman

Backmatter

Titel: A First Introduction to Quantum Computing and Information
verfasst von: Bernard Zygelman
Verlag: Springer International Publishing
Electronic ISBN: 978-3-031-66425-0
Print ISBN: 978-3-031-66424-3
DOI: https://doi.org/10.1007/978-3-031-66425-0