Using the ‘between quantum and classical ’ models that have been constructed explicitly within the hidden measurement approach of quantum mechanics we investigate the possibility to construct a ‘between quantum and classical ’ computer. In this view, the pure quantum computer and the classical Turing machine can be seen as two special cases of our general computer. We have shown in earlier research that the intermediate ‘between quantum and classical ’ systems cannot be described within standard quantum theory. We argue that the general categoral approach of state property systems might provide a unified framework for the study of these ‘between quantum and classical’ models, and hence also for the study of classical and quantum computers as special cases. 1
Using the ‘between quantum and classical ’ models that have been constructed explicitly within the hidden measurement approach of quantum mechanics we investigate the possibility to construct a ‘between quantum and classical ’ computer. In this view, the pure quantum computer and the classical Turing machine can be seen as two special cases of our general computer. We have shown in earlier research that the intermediate ‘between quantum and classical ’ systems cannot be described within standard quantum theory. We argue that the general categoral approach of state property systems might provide a unified framework for the study of these ‘between quantum and classical’ models, and hence also for the study of classical and quantum computers as special cases. 1
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.
It is reasonable to assume that quantum computations take place under the control of the classical world. For modelling this standard situation, we introduce a Classically-controlled Quantum Turing Machine (CQTM) which is a Turing machine with a quantum tape for acting on quantum data, and a classical transition function for a formalized classical control. In CQTM, unitary transformations and quantum measurements are allowed. We show that any classical Turing machine is simulated by a CQTM without loss of efficiency. Furthermore, we show that any k -tape CQTM is simulated by a 2-tape CQTM with a quadratic loss of efficiency. The gap between classical and quantum computations which was already pointed out in the framework of measurement-based quantum computation (see [S. Perdrix, Ph. Jorrand, Measurement-Based Quantum Turing Machines and their Universality, arXiv, quant-ph/0404146 , 2004]) is confirmed in the general case of classically-controlled quantum computation. In order to appreciate the similarity between programming classical Turing machines and programming CQTM, some examples of CQTM will be given in the full version of the paper. Proofs of lemmas and theorems are omitted in this extended abstract.
Contrary to the classical case, the relation between quantum programming languages and quantum Turing Machines (QTM) has not been fully investigated. In particular, there are features of QTMs that have not been exploited, a notable example being the intrinsic infinite nature of any quantum computation. In this paper, we propose a definition of QTM, which extends and unifies the notions of Deutsch and Bernstein & Vazirani. In particular, we allow both arbitrary quantum input, and meaningful superpositions of computations, where some of them are “terminated” with an “output”, while others are not. For some infinite computations an “output” is obtained as a limit of finite portions of the computation. We propose a natural and robust observation protocol for our QTMs, which does not modify the probability of the possible outcomes of the machines. Finally, we use QTMs to define a class of quantum computable functions—any such function is a mapping from a general quantum state to a probability distribution of natural numbers. We expect that our class of functions, when restricted to classical input-output, will not be different from the set of the recursive functions.
We study the observation of quantum Turing machines by allowing interactions between a quantum machine and its environment during the computation, whereas a quantum Turing machine --original model introduced by Deutsch-- remains isolated. We show that the introduction of observations leads to a weakening of the well formedness conditions of quantum Turing machines such that any (reversible or not) classical Turing machine is a special instance of a quantum machine. Moreover, observation of quantum Turing machines provides a formal solution to the halting process problem: the impossibility to know whether a given quantum Turing machine has actually reached its halting state. It also provides a more realistic abstract architecture of a quantum computer while most of the physical proposals of quantum computers are based on an hybrid classical-quantum architecture. However, we show that a natural formalisation of an observable quantum Turing machines leads to an over-powerful model solving undecidable problems. As a consequence, we introduce a restricted version of observable quantum Turing machines and we show that, under this restriction, any observable quantum Turing machine can be efficiently simulated by a quantum Turing machine. Finally, we discuss the potential application of the observable quantum Turing machine in the quest of a universal quantum Turing machine while recent papers have pointed out that a classical control is a key feature of a universal quantum machine.
Quantum computing is a promising approach of computation that is based on equations from Quantum Mechanics and nanotechnology. Understanding and simulating quantum operations on a classical computer has different approaches and is modeled in different ways. However, the most repute ways are Quantum Turing Machine, Network Model or circuit model of connected gates and Quantum Programming Language. Many simulators do exist that varies in many features such as implementation languages, target platforms, paradigms, visuality, and applied quantum algorithms. There was a developmental trade-off between the simplicity and the functionality of the simulator. The goals and research activities are outlined as follows: First, surveying the quantum computing world, its characteristics, issues, problems, and challenges. Second, surveying the existing quantum simulators, illustrating representative simulators of major approaches, their features, and functionalities. Third, introducing a framework to provide the power of quantum computing , while insisting the simplicity and high usability. Finally, providing a case study for a new quantum algorithm based on the introduced framework.
Quantum computing is a prolific research area, halfway between physics and computer science [27, 29, 52]. Most likely, its origins may be dated back to 70's, when some works on quantum information began to appear (see, e.g., [34, 37]). In early 80's, R.P. Feynman suggested that the computational power of quantum mechanical processes might be beyond that of traditional computation models [25]. Almost at the same time, P. Benioff already proved that such processes are at least as powerful as Turing machines [9]. In 1985, D. Deutsch [22] proposed the notion of a quantum Turing machine as a physically realizable model for a quantum computer. From the point of view of structural complexity, E. Bernstein and U. Vazirani introduced in [20] the class BQP of problems solvable in polynomial time on quantum Turing machines, focusing attention on relations with the corresponding deterministic and probabilistic classes P and BPP, respectively. Several works in the literature explored classical issues in complexity theory from the quantum paradigm perspective (see, e.g., [7, 60, 61]).