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Birkhäuser Basel eBooks

Quantum State Machines

Yorick Hardy, Willi-Hans Steeb

Published Jan 1, 2001• 0 citations
Publisher

Abstract

In this chapter we introduce the quantum state machine [80,119].The quantum state machine is an extension of classical finite state machines used to represent the computations possible in quantum computing. Quantum state machines introduce amplitudes for transitions between states to represent the parallelism available.
DOI
10.1007/978-3-0348-8366-5_19

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Very few papers have been written on the topic of a quantum version of the finite state machine, (or finite state automata). Furthermore, these papers only serve to define what a quantum finite state machine might be in the mathematical sense using the early languages of Turing machines. This paper seeks to further develop the notion of a quantum finite state machine (FSM) using constructs developed for the classical FSM and utilized for classical FSM design. In particular the quantum state transition diagram (QSTD) is constructed to further the understanding and realization of quantum finite state machines and quantum computers.
In quantum computers a unitary transformation is applied to a given initial state of a set of qubits through several quantum gates. Tests were conducted using Grover's algorithm and implemented in a variety of different ways on the publically available IBM quantum computers. Essentially, quantum computers manipulate quantum states instead of classical bits. Quantum information can be represented by the elementary units called quantum bits abbreviated as qubits or qbits. This chapter discusses the fundamental properties of quantum systems that are relevant to information processing. Quantum computation needs something much more than that quantum parallelism to be useful. This is achieved in Deutsch's algorithm which combines quantum parallelism with interference. One can simulate a classical circuit using a quantum circuit. That is, it is possible to perform classical computations on quantum computers. But the advantage of quantum computing is that powerful functions may be computed by making use of quantum parallelism, the fundamental feature of many quantum algorithms.
This introduction presents an overview of the key concepts discussed in the subsequent chapters of this book. The book introduces the motivation and presents and comment on recent related works in the literature about the quantum computing emulation by hardware and simulation by software. The conventional electronic computer is a machine that performs logical and arithmetic operations sequentially and/or in parallel on data coded according to binary logic. Quantum Computing has been the target of several researches due to its potential for increasing the speed of processing through the use of algorithms with intrinsic parallelism, pointing out to the possibility of polynomial time solutions for NP-Complete problems. The book describes the macro-architecture of the proposed quantum processor and its interaction with the conventional host wherein it is embedded. It defines quantum state measurement and propose an efficient micro-architecture for the unit that is responsible for the state observation of the quantum bits, thus allowing for quantum bit measurements.
Quantum computing is an emerging form of computing that uses the rules of quantum physics to process information in ways that are far different from how computers do it now. Quantum computing uses quantum bits, or qubits, which can exist in superpositions of states and get entangled with each other, which is different from classical computing, which uses a bit that represents binary values (0 or 1). These characteristics allow quantum devices to execute parallel computations, enabling significant speedups for specific categories of problems considered intractable for classical machines. This chapter gives a thorough introduction to quantum computing, starting with a historical overview of its foundations in quantum physics and significant theoretical advancements. It analyzes fundamental ideas including superposition, entanglement, quantum interference, and the no-cloning theorem, explaining how these elements distinguish quantum computation from traditional approaches. The advancement of quantum hardware is examined through several architectures, trapped ions, superconducting circuits, photonics, and topological qubits, together with software frameworks like Qiskit, Cirq, and PennyLane, which support algorithm development and simulation. The chapter examines fundamental quantum algorithms, such as Grover’s algorithm for database search, Shor’s algorithm for integer factorization, and hybrid variational methods designed for noisy intermediate-scale quantum devices. The practical applications in domains including as encryption, optimization, machine learning, chemistry, and finance emphasize the real-world significance of quantum technology. Challenges like decoherence, noise, scalability, and algorithmic maturity are rigorously analyzed, followed by an anticipated discourse on future trajectories including fault-tolerant architectures, quantum networking, and the development of quantum cloud services. This chapter presents a fundamental introduction for students, researchers, and practitioners seeking to comprehend the essential principles, functionalities, and possibilities of quantum computing in the post–Moore’s law period.
Computation and quantum information [1] are an extension of the information process concept, and they are used for the effects of quantum mechanic [2]. This new scheme provides a work scene that is not restricted to only two operations; on the contrary, this can gain multiple intermediate states as a result of these two ways of superposition. By performing this operation, the system can evaluate all possibilities in only one step by making a parallel computation; meanwhile, in the classic way, this evaluation is carried out independently of each other, and in different steps. The quantum parallelism can be defined as a reduction in time and an increase in the fast of information process. In this chapter, the fundamental elements that make possible to use the mechanical-quantum phenomena as a tool in the process information are presented.
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.