Quantum Computing for Engineers

Table of Contents

1. Frontmatter

2. Mathematical and Computational Preliminaries

   1. Frontmatter

   2. Chapter 1. Linear Algebra and Probability

      Osama M. Raisuddin, Suvranu De

      Abstract

      This chapter Presents the essential mathematical tools—linear algebra and probability—that form the backbone of quantum computation. Previous knowledge of elementary linear algebra is assumed. Dirac notation is adopted from the onset, and Pauli Matrices are introduced. Concepts relevant to this Book, including matrix norms, condition number, projection matrices, Hermitian matrices, unitary matrices, the eigenvalue decomposition, the singular value decomposition, and the Krylov subspace are presented with their key properties. The two Major classes of inear Ssystem solvers—direct and iterative solvers—are briefly introduced with a discussion of their asymptotic caling. The chapter concludes with an introduction to Kronecker products and their properties.

   3. Chapter 2. Polynomial Approximations

      Osama M. Raisuddin, Suvranu De

      Abstract

      This Chapter introduces techniques for approximating functions using polynomials, both at a point and over an interval, which are foundational for understanding several advanced quantum algorithms. The Taylor series expansion for approximating an analytical function around a point is presented with its error bounds around the approximation point. Chebyshev polynomials are presented as a route to approximating analytic functions with uniformly bounded error over an interval. Subsequently, evaluation of analytical functions of matrices is presented with the special case of diagonalizable matrices. Finally, we iscuss the matrix exponentiation problem—an elementary task in quantum computation—along with error bounds for approximate exponentiation using the Baker–Campbell–Hausdorff formula.

   4. Chapter 3. Theory of Computing

      Osama M. Raisuddin, Suvranu De

      Abstract

      This chapter introduces fundamental concepts from classical computation, including Turing machines and universal circuit families, along with examples of computable and incomputable problems. A brief introduction to automata is provided with increasing computational capabilities, starting with Boolean circuits and culminating with the all-powerful Turing machine and the limits of computable problems and solutions. The Turing machine is then generalized to universal, probabilistic, and non-deterministic Turing machines. A connection between the uniform computing model of Turing machines and Boolean circuits is then made using universal circuit families.

   5. Chapter 4. An Overview of Practical Classical Computing

      Osama M. Raisuddin, Suvranu De

      Abstract

      This chapter digresses from the theoretical study of computation to connect abstract models to hardware, covering transistors, logic gates, and the translation from high-level code to machine instructions. The discussion begins with the manifestation of Boolean logic gates as transistor circuits and progresses toward combinational circuits to add unsigned integers. This approach is shown to be unfeasible, and the discussion pivots toward sequential circuits. A clock signal is presented along with edge detectors and memory elements, which is shown to be a more scalable approach toward building memory and a general-purpose computer. The inner workings of a rudimentary CPU are introduced along with CPU instructions and layers of computer code abstraction. Finally, progress in classical computation is presented with a brief overview of modern computer architecture.

   6. Chapter 5. Information and Complexity Theory

      Osama M. Raisuddin, Suvranu De

      Abstract

      This chapter introduces classical complexity classes (P, NP, BPP) and the quantum class BQP, motivating quantum speedups while grounding expectations. The discussion begins with basic computational problems and the specific structure of decision problems. Problems are then categorized into classical complexity classes P, NP, and NP-hard, based on the conjectured computational power and resource requirements of the computational model that can solve them efficiently. Further complexity classes are defined, and the conjectured relation between classical and quantum complexity classes is presented. A concluding thought experiment links information theory to thermodynamics.

3. A Brief Introduction to Quantum Mechanics

   1. Frontmatter

   2. Chapter 6. A Gentle Introduction to Quantum Mechanics

      Osama M. Raisuddin, Suvranu De

      Abstract

      This chapter introduces the structure of quantum theory through foundational concepts, including quantum states, superposition, measurement, and probability amplitudes. The discussion starts with historical context leading to the birth of quantum mechanics. The six postulates of quantum mechanics are then formally presented. Emphasis is placed on developing intuition for how quantum systems behave differently from classical systems, with a minimal use of mathematical formalism.

   3. Chapter 7. The Stern–Gerlach Experiment

      Osama M. Raisuddin, Suvranu De

      Abstract

      This chapter presents the Stern–Gerlach experiment as a concrete illustration of measurement, state collapse, and spin quantization. The chapter illustrates how discrete measurement outcomes emerge in quantum systems and how they are related to the mathematical structure of quantum states. The Stern–Gerlach experiments are approached with a classical mechanics perspective; observational results are presented to test hypotheses, and the falsification of classical hypotheses is used to motivate the need for quantum mechanics. All the experiments are then modeled using quantum mechanics to demonstrate agreement.

   4. Chapter 8. Photon Polarization

      Osama M. Raisuddin, Suvranu De

      Abstract

      This chapter uses polarization states of photons to reinforce and generalize earlier concepts. The chapter explores basis changes and probabilistic outcomes in the context of light, providing an accessible experimental analogy for qubit operations. A series of experiments is presented for photons passing through polarizing filters with experimentally observed results. Photons and polarizing filters are then modeled using rotation matrices and quantum mechanics to demonstrate agreement with experimental results.

4. The Quantum Computing Model

   1. Frontmatter

   2. Chapter 9. Qubits, Quantum Registers, and Quantum Gates

      Osama M. Raisuddin, Suvranu De

      Abstract

      This chapter introduces qubits—the fundamental units of quantum information—and describes how multiple qubits combine into quantum registers. Essential one- and two-qubit quantum gates and their algebraic properties are presented, and their connection to standard bases for qubits is made. The representation of gates in quantum circuits is illustrated and the notion of universal quantum gate sets is presented with a statement of the Solovay–Kitaev theorem.

   3. Chapter 10. Quantum Measurements and Circuits

      Osama M. Raisuddin, Suvranu De

      Abstract

      This chapter covers quantum measurement theory, introducing measurement operators and the quantum circuit model. Measurement operators are presented as projection operators along with renormalization according to the Born rule with outputs being bitstrings measured in the computational basis. Quantum circuits are presented with an explanation of the sequence of operations and endianness of indexing. Finally, the principle of deferred measurement is presented as a transformation of hybrid classical-quantum circuits into equivalent quantum circuits.

   4. Chapter 11. Superposition and Entanglement

      Osama M. Raisuddin, Suvranu De

      Abstract

      This chapter explores the uniquely quantum concepts of superposition and entanglement using quantum circuits, emphasizing intuition and illustrative examples. The mathematical properties of entangled quantum states are presented, and example codes are provided to experimentally realize these quantum phenomena as minimal examples—uniform superposition and Bell states—on a quantum computer or quantum computer simulator.

   5. Chapter 12. Classical and Reversible Computation

      Osama M. Raisuddin, Suvranu De

      Abstract

      This chapter bridges classical computation and quantum logic, explaining classical logic embedding into quantum circuits and discussing the concept of reversible computation and quantum oracles. The chapter begins by demonstrating a scheme to embed classical Boolean circuits into quantum circuits, demonstrating that quantum computers can simulate classical computers using reversible operations. Bennett’s uncompute trick is then presented to efficiently simulate classical computers with a statement of the associated theorems. Finally, classical and quantum oracles are presented with a circuit transformation to convert basis state oracles into phase oracles.

   6. Chapter 13. Access Models and Data Representation

      Osama M. Raisuddin, Suvranu De

      Abstract

      This chapter introduces a collection of ideas necessary for the remainder of this book. Notable data-encoding schemes—basis encoding and amplitude encoding—are presented. Powerful access models for quantum operations—sparse access and block-encoding access—are defined with their polynomial equivalence. Useful linear algebra properties like the Hermitian dilation and the decomposition of operations in the Pauli basis are discussed with their potential pitfalls.

   7. Chapter 14. Limitations of Quantum Computers

      Osama M. Raisuddin, Suvranu De

      Abstract

      This chapter discusses fundamental theoretical limits on quantum computing, including key no-go theorems such as the no-cloning and no-deletion theorems, and limitations on quantum speedups. Key considerations like data transfer between classical and quantum computers and the restriction of unitary and measurement operations is analyzed to gain insight into the challenges associated with developing and utilizing quantum algorithms.

   8. Chapter 15. Simon’s, Deutsch–Jozsa, and Bernstein–Vazirani Algorithms

      Osama M. Raisuddin, Suvranu De

      Abstract

      This chapter provides concrete examples of quantum algorithms demonstrating exponential speedups compared to classical counterparts, emphasizing complexity class separations and the potential of quantum computing. Each algorithm is presented step-by-step to analyze the speedup. The Abelian Hidden Subgroup Problem is introduced as a unifying framework for exponential speedup, of which Simon’s problem is shown to be an instance.

5. Programming Quantum Computers

   1. Frontmatter

   2. Chapter 16. The Quantum Computing Stack

      Osama M. Raisuddin, Suvranu De

      Abstract

      This chapter introduces the conceptual structure of the quantum computing stack. It discusses hardware-level control, device-specific gate sets, dynamic circuit instructions, and the role of quantum assembly languages with examples, where appropriate, and the relations between these elements are discussed. It also provides an overview of error suppression, mitigation, and correction as they appear across the stack with examples of well-known techniques. A brief review of contemporary hardware performance metrics is presented with rule-of-thumb interpretations of key metrics.

   3. Chapter 17. Libraries for Quantum Computing

      Osama M. Raisuddin, Suvranu De

      Abstract

      This chapter surveys widely used frameworks and libraries for quantum software development. The chapter covers tools for algorithm prototyping, simulation, classical preprocessing, and cloud-based execution, with an emphasis on Qiskit and supporting libraries. The target applications and hardware for each library are emphasized, where appropriate.