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.
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.
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.
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 computing is emerging as a revolutionary paradigm in the field of computing, promising an exponential leap in computing power and problemsolving that challenges even the most advanced supercomputers of the classical era.At the heart of this revolution lies the peculiarity of the quantum world, where subatomic particles can exist in multiple states simultaneously, a phenomenon known as superposition, and entangle their states in a way that defies classical logic.It is this capacity for superposition and entanglement that gives quantum computing its disruptive power.One of the most intriguing properties of quantum computing is its ability to process information using Qubits, the quantum analogues of classical bits.While a classical bit can only be in a single 0 or 1 state, a Qubit can simultaneously represent a 0, a 1, or any superposition of these states, allowing multiple calculations to be performed in parallel.This massively parallel processing capability is the basis for the potential superiority of quantum computers over their classical counterparts in a wide range of applications.The architecture of quantum computers is inherently different from classical computers.While the latter use transistors to store and manipulate information in bits, quantum computers use Qubits, which can be implemented in a variety of physical systems, such as atoms, ion traps, photons, superconductors or nuclear magnetic resonance.The manipulation of the Qubits is carried out through quantum gates, fundamental operations that transform the quantum state of the Qubits and allow quantum calculations to be carried out.Furthermore, to maintain quantum coherence, the control and isolation of the quantum system from its environment is crucial, posing significant challenges in terms of designing and building scalable and robust quantum systems.
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.
The subject of quantum computing brings together ideas from classical information theory, computer science, and quantum physics. This review aims to summarize not just quantum computing, but the whole subject of quantum information theory. Information can be identified as the most general thing which must propagate from a cause to an effect. It therefore has a fundamentally important role in the science of physics. However, the mathematical treatment of information, especially information processing, is quite recent, dating from the mid-20th century. This has meant that the full significance of information as a basic concept in physics is only now being discovered. This is especially true in quantum mechanics. The theory of quantum information and computing puts this significance on a firm footing, and has led to some profound and exciting new insights into the natural world. Among these are the use of quantum states to permit the secure transmission of classical information (quantum cryptography), the use of quantum entanglement to permit reliable transmission of quantum states (teleportation), the possibility of preserving quantum coherence in the presence of irreversible noise processes (quantum error correction), and the use of controlled quantum evolution for efficient computation (quantum computation). The common theme of all these insights is the use of quantum entanglement as a computational resource. It turns out that information theory and quantum mechanics fit together very well. In order to explain their relationship, this review begins with an introduction to classical information theory and computer science, including Shannon's theorem, error correcting codes, Turing machines and computational complexity. The principles of quantum mechanics are then outlined, and the Einstein, Podolsky and Rosen (EPR) experiment described. The EPR-Bell correlations, and quantum entanglement in general, form the essential new ingredient which distinguishes quantum from classical information theory and, arguably, quantum from classical physics. Basic quantum information ideas are next outlined, including qubits and data compression, quantum gates, the `no cloning' property and teleportation. Quantum cryptography is briefly sketched. The universal quantum computer (QC) is described, based on the Church-Turing principle and a network model of computation. Algorithms for such a computer are discussed, especially those for finding the period of a function, and searching a random list. Such algorithms prove that a QC of sufficiently precise construction is not only fundamentally different from any computer which can only manipulate classical information, but can compute a small class of functions with greater efficiency. This implies that some important computational tasks are impossible for any device apart from a QC. To build a universal QC is well beyond the abilities of current technology. However, the principles of quantum information physics can be tested on smaller devices. The current experimental situation is reviewed, with emphasis on the linear ion trap, high- Q optical cavities, and nuclear magnetic resonance methods. These allow coherent control in a Hilbert space of eight dimensions (three qubits) and should be extendable up to a thousand or more dimensions (10 qubits). Among other things, these systems will allow the feasibility of quantum computing to be assessed. In fact such experiments are so difficult that it seemed likely until recently that a practically useful QC (requiring, say, 1000 qubits) was actually ruled out by considerations of experimental imprecision and the unavoidable coupling between any system and its environment. However, a further fundamental part of quantum information physics provides a solution to this impasse. This is quantum error correction (QEC). An introduction to QEC is provided. The evolution of the QC is restricted to a carefully chosen subspace of its Hilbert space. Errors are almost certain to cause a departure from this subspace. QEC provides a means to detect and undo such departures without upsetting the quantum computation. This achieves the apparently impossible, since the computation preserves quantum coherence even though during its course all the qubits in the computer will have relaxed spontaneously many times. The review concludes with an outline of the main features of quantum information physics and avenues for future research.
Today's computers work on bits that exist as either 0 or 1. Quantum computers aren't limited to two states; they encode information as quantum bits, or qubits, which can exist in superposition. Qubits represent atoms, ions, photons or electrons and their respective control devices that are working together to act as computer memory and a processor. Because a quantum computer can contain these multiple states simultaneously, it has the potential to be millions of times more powerful than today's most powerful supercomputers. A processor that can use registers of qubits will be able to perform calculations using all the possible values of the input registers simultaneously. This superposition causes a phenomenon called quantum parallelism, and is the motivating force behind the research being carried out in quantum computing. Due to technical obstacles, till date, a quantum computer has not yet been realized. But the concepts and ideas of quantum computing has been demonstrated using various methods like NMR, Ion Trap, Quantum Dot, Optical Methods, etc. A quantum computer manipulates qubits by executing a series of quantum gates, each a unitary transformation acting on a single qubit or pair of qubits. In applying these gates in succession, a quantum computer can perform a complicated unitary transformation to a set of qubits in some initial state. The qubits can then be measured, with this measurement serving as the final computational result. Research must devise a way to maintain decoherence and other potential sources of error at an acceptable level. Probably the most important idea in this field is the application of error correction in phase coherence as a means to extract information and reduce error in a quantum system without actually measuring that system. Thereby, quantum computers will emerge as the superior computational devices and perhaps one day make today's modern computer obsolete.