Using the formalism of the Boolean algebra, the process of computation in quantum computers is analyzed.The description of quantum bits -qubits -in the Hilbert space is given, quantum logic elements, quantum networks and the structure of an ideal quantum computer are considered.Peculiarities of quantum mechanical calculations related to the superposition principle, the interference of complex amplitudes and the existence of entangled states are discussed.
Using the formalism of the Boolean algebra, the process of computation in quantum computers is analyzed.The description of quantum bits -qubits -in the Hilbert space is given, quantum logic elements, quantum networks and the structure of an ideal quantum computer are considered.Peculiarities of quantum mechanical calculations related to the superposition principle, the interference of complex amplitudes and the existence of entangled states are discussed.
Summary form only given, as follows. Gives a system theoretic approach to the concept of quantum computing. A quantum computer is envisaged to be a system of quantum circuits, acting on a state space, which is a finite-say 2/sup n/-dimensional-complex Hilbert space. The circuitry is a sequence of unitary transforms U/sub t/ /spl epsiv/ SU(2/sup n/) followed by a measurement. These transforms, so-called quantum gates, are controlled by a classical computer, usually especially electromagnetic fields. The state space of a quantum computer has the structure of a Hermitian vector space. Thus it allows "simultaneous" superposition of orthogonal basis states (corresponding to classical states) with the possibility of constructive and destructive interference between different paths of computation. This principle allows the usage of so-called entangled states by preparing the superposition of special "bent" configurations of basis vectors in the product Hilbert space, e.g., as they are known from error control and cryptography. This latter "entanglement" not only promises to make a quantum computer much more powerful than a probabilistic one, but it also allows a method of parallelism. The reason for this lies not only in the fact that the 2/sup n/ dimensional Hilbert space is the n-fold tensor product of 2-dimensional spaces as it is given by spin-1/2-particles, like photons, representing a quantum bit. First of all, we shall address the fact that the power of quantum computing lies in the properties of entangled states as opposed to those of separable states which essentially would resemble classical circuits without giving the feature of exponential speed-up. We shall briefly describe possible physical realisations of quantum (optic, electronic, magnetic,...) devices allowing the engineering of Hamiltonians needed for quantum systems. Basic quantum gates, especially the quantum Horner-Toffoli gate will be discussed to display the principles and methodology for designing quantum circuits. An outlook towards the possible availability of nano- and mesoscopic technologies supporting this new architecture in future generations concludes based on a speculative worst/best case forecast of possible applications, e.g., in public key and conventional cryptography.
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.
This presentation provides a comprehensive overview of the foundational principles of quantum computation. It begins by contrasting classical and quantum information, highlighting how quantum bits (qubits) leverage superposition, entanglement, and interference to process information in ways that surpass classical limits. The document explains the four postulates of quantum mechanics: state space (qubits as unit vectors in a Hilbert space), time evolution (via unitary operators), measurement (probabilistic collapse of the state), and composite systems (tensor products of subsystems). Key concepts such as the Bloch sphere representation, quantum gates (e.g., Hadamard, CNOT, Pauli gates), and quantum circuits are discussed, illustrating how quantum operations manipulate qubit states. The presentation also covers quantum evolution, the gate model for computation, and the universality of certain gate sets, emphasizing the potential for quantum parallelism and the transformative impact of quantum computing on fields like cryptography and optimization.
Quantum mechanics has proven its position as the primary theory for accurately describing microscopic domains. Numerous fields in physics utilize quantum mechanics to great effects, with one prominent application being quantum computing. Quantum computing extends the field of quantum mechanics to information theory, striving to perform computational tasks utilizing quantum phenomena with notably greater efficiency compared to classical alternatives. In this thesis, I will go through the basics required in the field of quantum computing. To understand quantum computing demands adapting a new type of matrix mechanical view. This involves considering Hilbert spaces, which are linear vector space having a well defined inner product. Hilbert spaces contain state vectors describing quantum states, as well as operators for altering and measuring these states. The dynamics of quantum systems are governed by three quantum postulates. These specify Hilbert spaces and state vectors as the fundamental components of quantum systems, Hermitian operators as providing the physical manifestation of these systems, and finally, the Schrödinger's equation to describe the time evolution of such systems. These notions are required in producing quantum computing. Quantum computing uses qubits as the computational basis, analogous to classical bits but with the added capability of quantum superposition. Qubits are represented as two-state vectors, and can be physically realized with any quantum-behaving binary-valued system, like the electron spins. Qubits can be combined into higher dimensional states with the use of tensor products, causing the system's Hilbert space to expand as 2^n with n being the number of qubits, allowing for exponential computational speedup. Computational logic can be constructed by acting on the qubits using unitary operators, known as quantum gates. The combinations of qubits and quantum gates can be visualized as quantum circuits, depicted using specialized circuit diagram notations. An essential part of quantum computing, and the phenomena enabling exponential computational speedup, is quantum entanglement. Entanglement arises in interacting qubits, intertwining them so that the individual states become correlated. Entanglement can be characterized by being unable to reduce a combined state into its pure constituents. The amount of entanglement in a system can be measured using von Neumann entropy, analogous to Shannon entropy in classical information theory. Entanglement serves various purposes in quantum computing and is a defining feature distinguishing it from the classical domain. In this thesis, the entanglement properties of a periodic four-qubit circuit with alternating quantum gates were simulated. The simulation was performed computationally, and measured entanglement entropy as a function of the amount of circuit periods. The results demonstrated different characteristics of entanglement depending on the parameters used for the quantum gates, with the entanglement either being periodic, non-occurring or chaotic in nature.
Quantum computing is a process that incorporates interacting physical systems that represent quantum bits and quantum gates. We present the quantum bit (qubit), the quantum register and the quantum gates. The qubit is described as a vector in a two-dimensional Hilbert space and the quantum register, which comprises a number of qubits, as a vector in a multidimensional Hilbert space. Quantum gates are Hilbert space operators that rotate the qubit or the quantum register vectors. Quantum computations are modeled and described using a quantum circuit model. We also present a quantum computer simulator based on the circuit model of quantum computation. In this model quantum computations and quantum algorithms are represented by circuits, which comprise quantum gates and quantum registers. The well-known Deutsch's algorithm is described and the corresponding quantum circuit is presented. Possible applications of quantum computers are be presented and discussed.
Quantum computing stands as a pioneering field within computer science, leveraging the enigmatic principles of quantum mechanics to execute intricate calculations. At its heart lies the qubit, the elemental building block of quantum information, which resides in a state of superposition, representing both 0 and 1 concurrently. Quantum computers, constructed from assemblies of qubits, inhabit an expansive and intricate 2^n-dimensional complex Hilbert space, with their evolution governed by the Schrödinger equation. The process of quantum computation can be distilled into three key stages:1) Initialization of the input state: Initiating the quantum process; 2)Application of unitary transformations: Executing the desired transformations on the input state; 3) Measurement of the output state. Quantum computers serve as universal machines, capable of decomposing any unitary transformation into a sequence of fundamental gates. This chapter will reveal the importance of quantum mechanics, quantum computation and its applications.