Quantum computing allows us to solve some problems much faster than existing classical algorithms. Yet, the quantum computer has been believed to be no more powerful than the most general computing model—the Turing machine. Undecidable problems, such as the halting problem, and unrecognizable inputs, such as the real numbers, are beyond the theoretical limit of the Turing machine. I suggest a model for a quantum computer, which is less general than the Turing machine, but may solve the halting problem for any task programmable on it. Moreover, inputs unrecognizable by the Turing machine can be recognized by the model, thus breaking the theoretical limit for a computational task. A quantum computer is not just a successful design of the Turing machine as it is widely perceived now, but is a different, less general but more powerful model for computing, the practical realization of which may need different strategies than those in use now.
Quantum computing allows us to solve some problems much faster than existing classical algorithms. Yet, the quantum computer has been believed to be no more powerful than the most general computing model—the Turing machine. Undecidable problems, such as the halting problem, and unrecognizable inputs, such as the real numbers, are beyond the theoretical limit of the Turing machine. I suggest a model for a quantum computer, which is less general than the Turing machine, but may solve the halting problem for any task programmable on it. Moreover, inputs unrecognizable by the Turing machine can be recognized by the model, thus breaking the theoretical limit for a computational task. A quantum computer is not just a successful design of the Turing machine as it is widely perceived now, but is a different, less general but more powerful model for computing, the practical realization of which may need different strategies than those in use now.
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.
Abstract According to the Church-Turing Thesis, a Turing machine, and thus the programming languages and computers in existence, can simulate any reasonable computing device. However, there are physical devices that are exponentially more efficient than a Turing machine, and hence would prove exponentially difficult to simulate using such a machine. This chapter explores the concept of quantum computers and their potential not only to simulate quantum physics exponentially faster than classical computers but also to solve other problems exponentially faster. It considers algorithms that break cryptosystems, find needles in haystacks, and predict who can win a game. It first describes particles, waves, and amplitudes in quantum computation before turning to states and operators. It then examines how interference lets a quantum computer learn some things about a function, how quantum computers can solve factoring in polynomial time, the link between factoring and modern cryptography, graph isomorphism and the hidden subgroup problem, and quantum walks and scattering.
. We revisit the notion of a quantum Turing-machine , whose design is based on the laws of quantum mechanics. It turns out that such a machine is not more powerful, in the sense of computability, than the machine originally constructed by Turing. Quantum Turing-machines do not violate the Church–Turing thesis. The benefit of quantum computing lies in efficiency . Quantum computers appear to be more efficient, in time, than classical Turing-machines, however its exact additional computational power is unclear, as this question ties in with deep open problems in complexity theory. We will sketch where BQP, the quantum analogue of the complexity class P, resides in the realm of complexity classes. §1. Introduction . A decade before Turing developed his theory of computing, physicists struggled with the advent of quantum mechanics. During the famous 5th Solvay Conference in 1927 it was clear that a new era of physics had surfaced. Its strange features like superposition and entanglement still lead to heated discussions and much confusion. However strange and counter-intuitive, the theory has never been refuted by experiments that are performed daily and in great numbers throughout laboratories around the world. Time after time the predictions of quantum mechanics are in full agreement with experiment. Shortly after the advent of quantum mechanics, Church, Turing and Post developed the notion of computability [Chu36, Tur36, Pos36]. Less than 10 years later these formal ideas would be put to practice resulting in the ENIAC, the first general purpose machine.
Turing machines based on quantum logic can solve undecidable problems. In this paper we will give recursion-theoretical characterization of the computational power of this kind of quantum Turing machines. In detail, for the unsharp case, it is proved that Σ01 ∪Π01⊆ LTd(ε,Σ)(LTw(ε,Σ))⊆Π02 when the truth value lattice is locally finite and the operation ∧ is computable, where LTd(ε,Σ)(LTw(ε,Σ))denotes the class of quantum language accepted by these Turing machine in depth-first model (respectively, width-first model); for the sharp case, we can obtain similar results for usual orthomodular lattices.
A practical viewpoint links reality, representation, and language to calculation by the concept of Turing (1936) machine being the mathematical model of our computers. After the Gödel incompleteness theorems (1931) or the insolvability of the so-called halting problem (Turing 1936; Church 1936) as to a classical machine of Turing, one of the simplest hypotheses is completeness to be suggested for two ones. That is consistent with the provability of completeness by means of two independent Peano arithmetics discussed in Section I.Many modifications of Turing machines cum quantum ones are researched in Section II for the Halting problem and completeness, and the model of two independent Turing machines seems to generalize them.Then, that pair can be postulated as the formal definition of reality therefore being complete unlike any of them standalone, remaining incomplete without its complementary counterpart. Representation is formal defined as a one-to-one mapping between the two Turing machines, and the set of all those mappings can be considered as “language” therefore including metaphors as mappings different than representation. Section III investigates that formal relation of “reality”, “representation”, and “language” modeled by (at least two) Turing machines.The independence of (two) Turing machines is interpreted by means of game theory and especially of the Nash equilibrium in Section IV.Choice and information as the quantity of choices are involved. That approach seems to be equivalent to that based on set theory and the concept of actual infinity in mathematics and allowing of practical implementations.
The powerful idea of computation has accompanied the development of human civilization, has deeply changed the way we live and work, and has accelerated the advancement of many areas of sciences. In this article, we explore the power and limits of computation from several different perspectives. We will discuss topics from the models of computation and Church-Turing thesis, to the impact of the P versus NP problem and quantum computing on our understanding of the limits of computation. More concretely, we will explore the computability and the halting problem, the efficiency problem of computation, the P versus NP problem. We then move on to the discussion of quantum computation, quantum algorithm for factoring and its implications, quantum simulation and the relation between quantum and classical computations.