Quantum computation is, at bottom, a branch of applied algebra and functional analysis: the state of a quantum register is a unit vector in a finite-dimensional complex Hilbert space, computation is the action of the unitary group on that space, and the structure of the most important quantum algorithms is dictated by the representation theory of finite and continuous groups. This paper develops the mathematical scaffolding of the field from first principles and uses it to explain why quantum computers can outperform classical ones. We review complex inner-product spaces, self-adjoint and unitary operators, the spectral theorem, and the tensor-product construction that gives rise to entanglement and the exponential dimension of the state space. We then describe the qubit and its geometry, the group-theoretic classification of quantum gates (the special unitary groups, the Pauli and Clifford groups, and their Lie-algebraic generators), and the universality results that justify finite gate sets together with the Solovay–Kitaev theorem. With this apparatus we analyse the quantum Fourier transform, phase estimation, and the hidden subgroup problem, showing how Shor’s factoring algorithm, Grover’s search, and the quantum linear-systems algorithm arise as instances of a small number of algebraic templates. Finally we treat the stabilizer formalism of quantum error correction—where finite abelian groups over GF(2) play the central role—and the variational, near-term (NISQ) algorithms that dominate current practice. Throughout, the emphasis is on the algebraic invariants that make quantum speed-ups possible.
Quantum computation is, at bottom, a branch of applied algebra and functional analysis: the state of a quantum register is a unit vector in a finite-dimensional complex Hilbert space, computation is the action of the unitary group on that space, and the structure of the most important quantum algorithms is dictated by the representation theory of finite and continuous groups. This paper develops the mathematical scaffolding of the field from first principles and uses it to explain why quantum computers can outperform classical ones. We review complex inner-product spaces, self-adjoint and unitary operators, the spectral theorem, and the tensor-product construction that gives rise to entanglement and the exponential dimension of the state space. We then describe the qubit and its geometry, the group-theoretic classification of quantum gates (the special unitary groups, the Pauli and Clifford groups, and their Lie-algebraic generators), and the universality results that justify finite gate sets together with the Solovay–Kitaev theorem. With this apparatus we analyse the quantum Fourier transform, phase estimation, and the hidden subgroup problem, showing how Shor’s factoring algorithm, Grover’s search, and the quantum linear-systems algorithm arise as instances of a small number of algebraic templates. Finally we treat the stabilizer formalism of quantum error correction—where finite abelian groups over GF(2) play the central role—and the variational, near-term (NISQ) algorithms that dominate current practice. Throughout, the emphasis is on the algebraic invariants that make quantum speed-ups possible.
Hilbert spaces provide the fundamental mathematical framework for describing quantum mechanical systems. Their structure, characterized by an inner product and completeness, allows for the representation of quantum states as vectors and physical observables as self-adjoint operators. Key quantum phenomena such as superposition and entanglement find natural expression within this formalism. Superposition, where a quantum system can exist in multiple states simultaneously, is represented by linear combinations of basis vectors in the Hilbert space. Entanglement, a non-classical correlation between quantum systems, is described by non-separable state vectors in a tensor product of Hilbert spaces. These concepts are pivotal in quantum computing, where the unit of information, the qubit, is a two-level quantum system whose state is a vector in a two-dimensional complex Hilbert space (ℂ²). Quantum gates, which perform operations on qubits, are represented by unitary operators acting on these state vectors. The power of quantum computation, particularly in algorithms like Shor's or Grover's, stems from the ability to exploit superposition and entanglement, processes intrinsically described within the Hilbert space framework. Thus, a thorough understanding of Hilbert spaces is indispensable for grasping the principles of quantum mechanics and for advancing the field of quantum information and computation [1, 7]. The transition from classical bits to quantum qubits, and from classical logic gates to quantum unitary operations, is entirely predicated on the mathematical properties endowed by Hilbert spaces.
Quantum computing represents a transformative paradigm that leverages the principles of quantum mechanics, such as superposition and entanglement, to perform computations beyond the capabilities of classical systems. The development of efficient quantum algorithms and architectures relies heavily on robust mathematical frameworks, among which group theory plays a fundamental role. Group-theoretic structures provide a systematic way to describe symmetries, unitary transformations, and state evolution in quantum systems. In particular, unitary and special unitary groups govern the behavior of quantum gates, while representation theory enables the analysis of multi-qubit operations and composite systems. Moreover, group theory underpins several key aspects of quantum computing, including the design of quantum algorithms, error correction schemes, and cryptographic protocols. Notable algorithms such as Shor’s Algorithm and Grover’s Algorithm exploit algebraic structures to achieve computational advantages. This review aims to provide a comprehensive overview of the interplay between group theory and quantum computing, highlighting fundamental concepts, major applications, and recent developments. The paper also identifies current challenges and outlines future research directions, thereby offering a consolidated resource for researchers in mathematics, physics, and quantum information science.
(Abridged abstract.) In this thesis we introduce new models of quantum computation to study the emergence of quantum speed-up in quantum computer algorithms. Our first contribution is a formalism of restricted quantum operations, named normalizer circuit formalism, based on algebraic extensions of the qubit Clifford gates (CNOT, Hadamard and π/4-phase gates): a normalizer circuit consists of quantum Fourier transforms (QFTs), automorphism gates and quadratic phase gates associated to a set G, which is either an abelian group or abelian hypergroup. Though Clifford circuits are efficiently classically simulable, we show that normalizer circuit models encompass Shor's celebrated factoring algorithm and the quantum algorithms for abelian Hidden Subgroup Problems. We develop classical-simulation techniques to characterize under which scenarios normalizer circuits provide quantum speed-ups. Finally, we devise new quantum algorithms for finding hidden hyperstructures. The results offer new insights into the source of quantum speed-ups for several algebraic problems. Our second contribution is an algebraic (group- and hypergroup-theoretic) framework for describing quantum many-body states and classically simulating quantum circuits. Our framework extends Gottesman's Pauli Stabilizer Formalism (PSF), wherein quantum states are written as joint eigenspaces of stabilizer groups of commuting Pauli operators: while the PSF is valid for qubit/qudit systems, our formalism can be applied to discrete- and continuous-variable systems, hybrid settings, and anyonic systems. These results enlarge the known families of quantum processes that can be efficiently classically simulated. This thesis also establishes a precise connection between Shor's quantum algorithm and the stabilizer formalism, revealing a common mathematical structure in several quantum speed-ups and error-correcting codes.
Quantum computing represents a fundamentally new paradigm in information processing, in which information is encoded and manipulated using two-level quantum systems known as qubits, whose evolution is governed by the principles of quantum mechanics. This chapter presents a structured introduction to the theoretical foundations of quantum computation, beginning with the limitations of classical computation and the physical constraints that motivate the need for a quantum model of computation. We develop the mathematical framework required to describe quantum information, introducing qubits as state vectors in Hilbert space and examining the roles of superposition, measurement, and unitary evolution. Single-qubit operations are presented through the Pauli and Hadamard gates, along with their geometric interpretation on the Bloch sphere. Building on this formalism, the discussion is extended to multi-qubit systems using tensor products, and quantum entanglement is introduced as a non-classical correlation fundamental to quantum information processing. The chapter then explores quantum algorithms, focusing on Grover’s unstructured search algorithm, to demonstrate how superposition, interference, and entanglement can be harnessed to achieve computational advantages over classical approaches. Finally, the chapter discusses applications of quantum computation, including quantum simulation, cryptography, search and optimization problems, and near-term hybrid quantum algorithms such as the Variational Quantum Eigensolver (VQE)and the Quantum Approximate Optimization (QAO) Algorithm, highlighting their relevance in the noisy intermediate-scale quantum (NISQ) era.
Quantum computing uses a mathematical language that can seem unfamiliar to readers from programming, engineering and other technical backgrounds. Qubits are represented by vectors, quantum gates by matrices, measurement outcomes by eigenvalues, and probability amplitudes by complex numbers. Quantum Computing Preliminaries develops this language carefully, connecting each mathematical idea with its computational and physical meaning. The book begins with complex vector spaces and explains why quantum states require more than classical probabilities. It introduces qubits, superposition, normalisation, basis vectors and Dirac bra–ket notation, supported by worked calculations and geometric interpretation using the Bloch sphere. The discussion then turns to inner products and the geometry of quantum states. Readers learn how to calculate norms, test orthogonality, determine state overlap and apply the Born rule. Particular attention is given to complex conjugation and the distinction between the ordinary dot product and the Hermitian inner product used in quantum mechanics.The treatment of matrix operators establishes the roles of linear, unitary and Hermitian operators. It explains matrix multiplication, non-commutativity, eigenvalues, eigenvectors, reversibility and norm preservation. The Pauli and Hadamard gates provide concrete examples that link matrix algebra to rotations, the Bloch sphere, quantum-state evolution, and measurement. Outer products and projection operators are developed as tools for constructing operators, selecting state components and describing ideal projective measurements. The book examines idempotence, completeness, repeated measurement and the distinction between the ket–bra outer product and the tensor product used to form composite quantum systems. The final chapter explores changes of basis, operator representations and diagonalisation. It shows how the same physical state or operation can have different numerical representations in the computational and Hadamard bases. Readers derive the similarity transformation, use outer-product substitution, diagonalise Hermitian operators and interpret the spectral decomposition geometrically. Examples involving the Pauli-X, Pauli-Y, Pauli-Z and Hadamard operators demonstrate how basis choice exposes phase, population and measurement structure without changing the underlying physics. Throughout the book, Python and NumPy examples translate the mathematics into executable calculations. The programs demonstrate complex-vector operations, inner products, eigendecomposition, basis transformation and numerical verification while explaining the importance of using operations suited to complex Hilbert spaces. Each chapter includes conceptual, mathematical, computational and challenge exercises. Detailed solutions provide step-by-step derivations, program output and physical interpretation, allowing the book to support both guided study and independent practice. Quantum Computing Preliminaries is intended for students, software developers, engineers and technically minded readers preparing to study quantum circuits, algorithms, simulation, measurement, entanglement or error correction. It assumes familiarity with elementary algebra and programming but does not require prior training in quantum mechanics. Its purpose is to provide a clear mathematical foundation for understanding how quantum information is represented, transformed and measured.
The Quantum Fourier Transform (QFT)andQuantumPhaseEstimation (QPE)are twofundamental algorithmic frameworks in quantum computation, and they provide the mathematical basis of many exponential-speedup quan tum algorithms, such as Shor’s factoring algorithm and quantum simulation techniques. In this work we present a detailed theoretical and computational study of the Quantum Fourier Transform and its integration in the Quan tum Phase Estimation. The discussion begins with the linear algebraic structure of the Hilbert spaces, the unitary transformations, and the complex vector representations that enable quantum interference and superposition. The QFT is studied as the unitary transform of computational basis states in the amplitudes in the frequency domain by phase rotations and tensor-product decompositions. The efficient circuit implementation in terms of Hadamard and controlled phase gates of this transform is discussed. The advantage of the transform in terms of polynomial time complexity over the classical discrete Fourier transform in some quantum scenarios is discussed.The QFT is implemented and the Quantum Phase Estimation algorithm is proposed to obtain the eigenphase information of the unitary operators. In the report the mathematical representation of QPE is derived by using the controlled unitary evolution, the inverse Fourier transforms and the measurement theory in finite dimensional Hilbert spaces. We discuss the probabilistic interpretation of phase extraction in detail and the convergence properties of the bi nary phase approximation. We also discuss the relation of QPE to eigenvalue problems in quantum mechanics and its applications to Hamiltonian simulation, quantum chemistry, cryptography, and hidden subgroup prob lems.Implementation aspects of these algorithms are studied using quantum circuit representations and oracle based computation models. We also discuss error sensitivity, qubit precision requirements, decoherence limita tions and practical constraints of Noisy Intermediate-Scale Quantum (NISQ) hardware. The work synthesizes the mathematical foundations, computational complexity analysis and physical realizations of the QFT and the QPE, demonstrating how the Fourier-based quantum transformations provide the essential mechanism underlying modern quantum algorithmic speedups and scalable quantum information processing.