Quantum Turing Machines Computations and Measurements

Contrary to the classical case, the relation between quantum programming languages and quantum Turing Machines (QTM) has not being 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 and 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, that 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 be not different from the set of the recursive functions.Comment: arXiv admin note: substantial text overlap with arXiv:1504.02817 To appear on MDPI Applied Sciences, 202

Quantum Simulations of Classical Random Walks and Undirected Graph Connectivity

AbstractWhile it is straightforward to simulate a very general class of random processes space-efficiently by non-unitary quantum computations (e.g., quantum computations that allow intermediate measurements to occur), it is not currently known to what extent restricting quantum computations to be unitary affects the space required for such simulations. This paper presents a method by which a limited class of random processes—random walks on undirected graphs—can be simulated by unitary quantum computations in a space-efficient (and time-efficient) manner. By means of such simulations, it is demonstrated that the undirected graph connectivity problem for regular graphs can be solved by one-sided error quantum Turing machines that run in logspace and require a single measurement at the end of their computations. It follows that symmetric logspace is contained in a quantum analogue of randomized logspace that disallows intermediate measurements

Measurement-Based Quantum Turing Machines and their Universality

Quantum measurement is universal for quantum computation. This universality allows alternative schemes to the traditional three-step organisation of quantum computation: initial state preparation, unitary transformation, measurement. In order to formalize these other forms of computation, while pointing out the role and the necessity of classical control in measurement-based computation, and for establishing a new upper bound of the minimal resources needed to quantum universality, a formal model is introduced by means of Measurement-based Quantum Turing Machines.Comment: 13 pages, based upon quant-ph/0402156 with significant improvement

Measurement-Based Quantum Turing Machines and Questions of Universalities

Quantum measurement is universal for quantum computation. This universality allows alternative schemes to the traditional three-step organisation of quantum computation: initial state preparation, unitary transformation, measurement. In order to formalize these other forms of computation, while pointing out the role and the necessity of classical control in measurement-based computation, and for establishing a new upper bound of the minimal resources needed to quantum universality, a formal model is introduced by means of Measurement-based Quantum Turing Machines.Comment: 12 pages, 9 figure

Classically-Controlled Quantum Computation

Quantum computations usually take place under the control of the classical world. We introduce a Classically-controlled Quantum Turing Machine (CQTM) which is a Turing Machine (TM) with a quantum tape for acting on quantum data, and a classical transition function for a formalized classical control. In CQTM, unitary transformations and measurements are allowed. We show that any classical TM is simulated by a CQTM without loss of efficiency. The gap between classical and quantum computations, already pointed out in the framework of measurement-based quantum computation is confirmed. To appreciate the similarity of programming classical TM and CQTM, examples are given.Comment: 20 page

Quantum Random Self-Modifiable Computation

Among the fundamental questions in computer science, at least two have a deep impact on mathematics. What can computation compute? How many steps does a computation require to solve an instance of the 3-SAT problem? Our work addresses the first question, by introducing a new model called the ex-machine. The ex-machine executes Turing machine instructions and two special types of instructions. Quantum random instructions are physically realizable with a quantum random number generator. Meta instructions can add new states and add new instructions to the ex-machine. A countable set of ex-machines is constructed, each with a finite number of states and instructions; each ex-machine can compute a Turing incomputable language, whenever the quantum randomness measurements behave like unbiased Bernoulli trials. In 1936, Alan Turing posed the halting problem for Turing machines and proved that this problem is unsolvable for Turing machines. Consider an enumeration E_a(i) = (M_i, T_i) of all Turing machines M_i and initial tapes T_i. Does there exist an ex-machine X that has at least one evolutionary path X --> X_1 --> X_2 --> . . . --> X_m, so at the mth stage ex-machine X_m can correctly determine for 0 <= i <= m whether M_i's execution on tape T_i eventually halts? We demonstrate an ex-machine Q(x) that has one such evolutionary path. The existence of this evolutionary path suggests that David Hilbert was not misguided to propose in 1900 that mathematicians search for finite processes to help construct mathematical proofs. Our refinement is that we cannot use a fixed computer program that behaves according to a fixed set of mechanical rules. We must pursue methods that exploit randomness and self-modification so that the complexity of the program can increase as it computes.Comment: 50 pages, 3 figure

Quantum Robots and Quantum Computers

Validation of a presumably universal theory, such as quantum mechanics, requires a quantum mechanical description of systems that carry out theoretical calculations and experiments. The description of quantum computers is under active development. No description of systems to carry out experiments has been given. A small step in this direction is taken here by giving a description of quantum robots as mobile systems with on board quantum computers that interact with environments. Some properties of these systems are discussed. A specific model based on the literature descriptions of quantum Turing machines is presented.Comment: 18 pages, RevTex, one postscript figure. Paper considerably revised and enlarged. submitted to Phys. Rev.