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2021 | Book

Foundations of Quantum Mechanics Read chapter Read first chapter

Fundamentals of Quantum Computing

Theory and Practice

Author: Venkateswaran Kasirajan

Publisher: Springer International Publishing

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This introductory book on quantum computing includes an emphasis on the development of algorithms. Appropriate for both university students as well as software developers interested in programming a quantum computer, this practical approach to modern quantum computing takes the reader through the required background and up to the latest developments.

Beginning with introductory chapters on the required math and quantum mechanics, Fundamentals of Quantum Computing proceeds to describe four leading qubit modalities and explains the core principles of quantum computing in detail. Providing a step-by-step derivation of math and source code, some of the well-known quantum algorithms are explained in simple ways so the reader can try them either on IBM Q or Microsoft QDK. The book also includes a chapter on adiabatic quantum computing and modern concepts such as topological quantum computing and surface codes.

Features:

o Foundational chapters that build the necessary background on math and quantum mechanics.

o Examples and illustrations throughout provide a practical approach to quantum programming with end-of-chapter exercises.

o Detailed treatment on four leading qubit modalities -- trapped-ion, superconducting transmons, topological qubits, and quantum dots -- teaches how qubits work so that readers can understand how quantum computers work under the hood and devise efficient algorithms and error correction codes. Also introduces protected qubits - 0-π qubits, fluxon parity protected qubits, and charge-parity protected qubits.

o Principles of quantum computing, such as quantum superposition principle, quantum entanglement, quantum teleportation, no-cloning theorem, quantum parallelism, and quantum interference are explained in detail.

A dedicated chapter on quantum algorithm explores both oracle-based, and Quantum Fourier Transform-based algorithms in detail with step-by-step math and working code that runs on IBM QisKit and Microsoft QDK. Topics on EPR Paradox, Quantum Key Distribution protocols, Density Matrix formalism, and Stabilizer formalism are intriguing. While focusing on the universal gate model of quantum computing, this book also introduces adiabatic quantum computing and quantum annealing.

This book includes a section on fault-tolerant quantum computing to make the discussions complete. The topics on Quantum Error Correction, Surface codes such as Toric code and Planar code, and protected qubits help explain how fault tolerance can be built at the system level.

Table of Contents

Frontmatter

Foundations

Frontmatter
Chapter 1. Foundations of Quantum Mechanics
Abstract
The late 19th century and the beginning of the 20th century were buzzing with several scientific advancements. The invention of subatomic particles, the birth of the theory of relativity, and the formulation of quantum mechanics were some of the most outstanding achievements of this period that changed the way we understand nature. This chapter takes the reader to this era to touch upon how nature works and introduces wave-particle duality, quantization, wavefunctions, probability amplitudes, quantum measurements, and exotic states of matter. A quick read of this chapter shall help readers grasp advanced topics coming up in the forthcoming chapters.
Venkateswaran Kasirajan
Chapter 2. Dirac’s Bra-ket Notation and Hermitian Operators
Abstract
This chapter is a refresher course on linear algebra and probability theory and starts from the basic definitions of vectors and matrices. After the introduction of linear vector spaces, this chapter teaches how to use Dirac's bra-ket notation. Concepts of expectation values, eigenstates, eigenvalues, and eigenfunctions are explained with examples. Sections on tensor math, statistics, and probability will be of help in forthcoming chapters.
Venkateswaran Kasirajan
Chapter 3. The Quantum Superposition Principle and Bloch Sphere Representation
Abstract
This chapter introduces Hilbert space, Schrödinger wave equation, quantum superposition principle and explains the postulates of quantum mechanics. Quantum tunneling is explained as an application of the Schrödinger wave equation. The concept of a quantum two-state system (qubits) is developed with the help of the Stern and Gerlach experiment. Bloch sphere representation is used to explain the gate operations on the qubits. This chapter discusses the physics of quantum measurements and concludes with an introduction to qudits and qutrits.
Venkateswaran Kasirajan

Essentials

Frontmatter
Chapter 4. Qubit Modalities
Abstract
The goal of this chapter is to introduce the current generation of Noisy Intermediate Scale Quantum Computers. This chapter begins by explaining the everyday vocabulary of quantum computing and qubit metrics. After these introductions, qubit physics of four leading qubit modalities - trapped ion, superconducting transmons, topological qubits, and quantum dots are described in detail. This chapter concludes with a note on the functionality of dilution refrigerators.
Venkateswaran Kasirajan
Chapter 5. Quantum Circuits and DiVincenzo Criteria
Abstract
 The first part of this chapter guides the readers to set up their systems for quantum computing with the necessary instructions to install IBM Quantum Experience and Microsoft QDK. After this setup, the readers are given a walkthrough of the gate model of universal quantum computing and building simple quantum circuits. The second part of this chapter introduces core concepts such as quantum parallelism, quantum interference, quantum entanglement, no-cloning theorem, quantum teleportation, and super dense coding. Gottesman-Knill theorem, GHZ states, Walsh-Hadamard transform, phase kickback, and partial measurement are some topics included in this chapter. This chapter concludes with a discussion on DiVincenzo's criteria for quantum computing.
Venkateswaran Kasirajan
Chapter 6. Quantum Communications
Abstract
Quantum Information Theory is a growing subject of specialization since quantum communication has a good potential for the industry to adopt. This chapter is a short introduction to some of the basic concepts of this focus area and begins with a brief introduction to the Einstein–Podolsky–Rosen (EPR) Paradox. Measurement of the Bell states, explanation of the local realism, and proof of Clauser, Horne, Shimony, and Holt (CHSH) inequality are the first few topics explored. The concept of density matrices, differences between pure and mixed states, a procedure for validating entanglement, and the no-communication theorem are introduced next. The final sections of this chapter introduce quantum communications using entangled photons, Quantum Key Distribution protocols, and RSA security.
Venkateswaran Kasirajan
Chapter 7. Quantum Algorithms
Abstract
  This chapter explores some of the fundamental quantum algorithms by starting with a simple ripple adder circuit, an example that helps readers build layered quantum circuits and apply unitary transformations in each stage—qubit by qubit, and evolve the final quantum state.After completing this exercise, this chapter briefs quantum Fourier transform (QFT) and applies it to perform arithmetic. Deutsch–Jozsa algorithm, Bernstein–Vazirani problem, and Simon’s algorithm are some of the oracle-based algorithms discussed, followed by a detailed discussion on Grover's search, modular exponentiation, and Shor's algorithm.A quantum version of the k-means algorithm, quantum phase estimation and HHL algorithm that solves linear equations are interesting methods discussed in the last few sections.Throughout this chapter, the theory is mixed with a working code. Step-by-step derivation of the math is provided to help improve understanding. This chapter concludes with a short note on the complexity theory.
Venkateswaran Kasirajan
Chapter 8. Adiabatic Optimization and Quantum AnnealingQuantum Annealing
Abstract
Adiabatic quantum computing (AQC) is an alternate form of quantum computation that is based on the quantum adiabatic theorem. A related concept is quantum annealing. In an AQC based computation, an initial Hamiltonian in its ground state is evolved slowly to a final Hamiltonian whose ground state encodes the solution to the problem. AQC is used in solving optimization problems and developing heuristic algorithms. This introductory chapter describes these methods.
Venkateswaran Kasirajan
Chapter 9. Quantum Error Correction
Abstract
Building from the classical Shannon's entropy, this chapter introduces Pauli Group and quantum error correction mechanisms in correcting bit-flip, phase-flip, and mixed errors that can typically occur in quantum circuits. After building the theoretical framework, this chapter introduces the Gottesman-Knill theorem and the stabilizer formalism. The stabilizer formalism is then applied to Shor's 9-qubit error correction code, CSS code, and Steane's 7-qubit error correction code.While discussing the path towards fault-tolerant quantum systems, this chapter explores ways to solve measurement errors and introduces the toric code and the surface code.Protected qubits such as 0-π qubit, Fluxon-parity protected superconducting qubit, and parity protected superconducting-semiconductor qubit discussed in the closing sections of this chapter explain to the readers how qubits can be protected from errors at the physical level.
Venkateswaran Kasirajan
Chapter 10. Conclusion
Abstract
We have witnessed many technological revolutions, and we know that classical computing took more than seven decades to reach the current level. The first transistor was about our palm’s size, and we did not build a supercomputer with that. The first commercial device that rolled out with the transistor was a hearing aid. With quantum computing, we are somewhat in an early transistor-like period. The good thing is that we a handful of well-established quantum algorithms, and there are quantum devices currently available for experimentation.However, to the inquiring minds, there are some unanswered questions. The top three questions that come to our mind are—How many qubits do we need? Can fault-tolerant quantum computers of commercial-grade be ever built? What does the future hold? This concluding chapter tries to answer these frequently asked questions and sets up the directions for the readers.
Venkateswaran Kasirajan
Backmatter
Metadata
Title
Fundamentals of Quantum Computing
Author
Venkateswaran Kasirajan
Copyright Year
2021
Electronic ISBN
978-3-030-63689-0
Print ISBN
978-3-030-63688-3
DOI
https://doi.org/10.1007/978-3-030-63689-0

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