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CRC Press eBooks

Basic ideas of classical and quantum computational models and complexity classes

Anirban Pathak

Published Jun 20, 2013• 0 citations
Publisher

Abstract

In this chapter we briefly describe the basic ideas of classical and quantum computational models. To be precise, we will provide an overview of the classical (irreversible, probabilistic and reversible) and quantum Turing machines, the need for error correction and the quantum circuit model of computation. We also describe a few complexity classes that are relevant for the study of quantum computing and quantum cryptography.

DOI
10.1201/b15007-6

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In this chapter we briefly describe the basic ideas of classical and quantum computational models. To be precise, we will provide an overview of the classical (irreversible, probabilistic and reversible) and quantum Turing machines, the need for error correction and the quantum circuit model of computation. We also describe a few complexity classes that are relevant for the study of quantum computing and quantum cryptography.
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Now that we have the laws for qubits, we need to develop a system for meaningfully manipulating them. Much of the current paradigm for quantum computing is motivated by classical computation theory, especially the circuit model for computation. In this chapter, we will briefly overview the classical model of Boolean circuit theory, and also some of the theoretical concepts involved in classifying the computational complexity of problems. Roger Penrose [53] has a beautiful account of the history of the theory of computation. Another wonderful book on similar lines is that of Douglas Hofstadter [41], both of which will stimulate you to think along the lines of a computer scientist.
Lots of efforts in the last decades have been done to prove or disprove whether the set of polynomially bounded problems is equal to the set of polynomially verifiable problems. This paper will present an overview of the current beliefs of quantum complexity theorists and discussion detail the impacts these beliefs may have on the future of the field. We introduce a new form of Turing machine based on the concepts of quantum mechanics and investigate the domain of this novel definition for computation. The main object is to show the domain of these computation machines and the new definition of algorithms using these automata. In section one we give a brief fundamental overview of classical complexity theory, Turing machine and all the fundamental concepts required for the next chapters. In section two we describe various classes of classical complexity automata. In next chapter we introduce quantum complexity featuring quantum Turing machines and discuss its relationship to classical complexity. Section four presents a relationship between quantum complexity classes based on the quantum Turing machines and quantum oracles, and their relationship with corresponding classical complexity classes. And section five presents a discussion on the practical phenomena in problem solving using quantum computers. Finally in section five we speculate briefly on the direction we believe the field to be headed, and what might reasonably be expected in the future
The physics and the mathematics of computation are examined to provide a foundation and perspective for the investigation of the quantum mechanics of computation. Our purpose is to explore the fundamental limits and constraints imposed on computation by Nature through the laws of physics and the mathematics of computational complexity. Inasmuch as information storage and transmission are an integral part of computation, their physical bounds are considered. The computer is viewed both physically and mathematically as a dynamical system, and is depicted in terms of the basic Turing machine paradigm. Three fundamental classes of the Turing machine are defined; the deterministic, stochastic and quantum Turing machines. Hamiltonian models and physical realizations of quantum computing are described. Quantum computers can perform some tasks which have no classical analogue, but they cannot compute functions that are non-computable by classical means. Some classically intractable problems can be solved with quantum computers.
There is strong evidence that computers based upon the principles of quantum physics represent an inherently new and more powerful model of computation. Such computers violate the modern form of the Church-Turing thesis (which lies at the foundations of computer science). This thesis can be informally summarized as follows: all physical implementations of computing devices can be described by the same abstract model of computation-the probabilistic Turing Machine or the equivalent random access machine. In this paper, we will describe two formal models for quantum computers: quantum circuits and quantum Turing Machines, introduced by Deutsch [1985].
Models of quantum computation and quantum programming languages The goal of the presented paper is to provide an introduction to the basic computational models used in quantum information theory. We review various models of quantum Turing machine, quantum circuits and quantum random access machine (QRAM) along with their classical counterparts. We also provide an introduction to quantum programming languages, which are developed using the QRAM model. We review the syntax of several existing quantum programming languages and discuss their features and limitations.