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We explore in the framework of Quantum Computation the notion of Computabil-ity, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm for Hilbert's tenth problem, which is equivalent to the Turing halting problem and is known to be mathematically noncomputable, is proposed where quantum continuous variables and… (More)

We employ quantum mechanical principles in the computability exploration of the class of classically noncom-putable Hilbert's tenth problem which is equivalent to the Turing halting problem in Computer Science. The Quantum Adiabatic Theorem enables us to establish a connection between the solution for this class of problems and the asymptotic behaviour of… (More)

We explore the possibility of using quantum mechanical principles for hypercomputation through the consideration of a quantum algorithm for computing the Turing halting problem. The mathematical noncomputability is compensated by the measurability of the values of quantum ob-servables and of the probability distributions for these values. Some previous… (More)

We explore in the framework of Quantum Computation the notion of computabil-ity, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm that exploits the quantum adiabatic which is equivalent to the Turing halting problem and known to be mathematically noncomputable. Gener-alised quantum algorithms are also… (More)

Despite the recursive non-computability of Hilbert's tenth problem, we outline and argue for a quantum algorithm that is based on the Quantum Adiabatic Theorem. It is explained how this algorithm can solve Hilbert's tenth problem. The algorithm is then considered in the context of several " no-go " arguments against such hypercomputation. Logical arguments… (More)

We show how to determine the k-th bit of Chaitin's algorithmically random real number, Ω, by solving k instances of the halting problem. From this we then reduce the problem of determining the k-th bit of Ω to determining whether a certain Diophantine equation with two parameters, k and N , has solutions for an odd or an even number of values of N. We also… (More)

- Tien D. Kieu
- IJUC
- 2009

- Toby Ord, Tien D. Kieu
- IJUC
- 2009

While it is well known that a Turing machine equipped with the ability to ip a fair coin cannot compute more that a standard Turing machine, we show that this is not true for a biased coin. Indeed, any oracle set X may be coded as a probability p X such that if a Turing machine is given a coin which lands heads with probability p X it can compute any… (More)

- Tien D. Kieu
- CCA
- 2005

Inspired by Quantum Mechanics, we reformulate Hilbert's tenth problem in the domain of integer arithmetics into problems involving either a set of infinitely-coupled non-linear differential equations or a class of linear Schrödinger equations with some appropriate time-dependent Hamiltonians. We then raise the questions whether these two classes of… (More)

We show how to determine the k-th bit of Chaitin's algorithmically random real number Ω by solving k instances of the halting problem. From this we then reduce the problem of determining the k-th bit of Ω to determining whether a certain Diophantine equation with two parameters, k and N , has solutions for an odd or an even number of values of N. We also… (More)

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