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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 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
- 2006

Quantum computation has attracted much attention and investment lately through its theoretical potential to speed up some important computations as compared to classical Turing computation. The most well-known and widely-studied form of quantum computation is the (standard) model of quantum circuits [1] which comprise several unitary quantum gates… (More)

- Tien D Kieu
- 2003

We propose a quantum algorithm for the classically non-computable Hilbert's tenth problem, which ultimately links to the Turing halting problem. Quantum continuous variables and quantum adiabatic evolution are employed for an implementation. Also discussed are a method for the time estimation for the adiabatic evolution, and a comparison with more the… (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)

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)

- Toby Ord, Tien D. Kieu
- 2005

The diagonal method is often used to show that Turing machines cannot solve their own halting problem. There have been several recent attempts to show that this method also exposes either contradiction or arbitrariness in other theoretical models of computation which claim to be able to solve the halting problem for Turing machines. We show that such… (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)

- Tien D Kieu
- 2001

Inspired by Quantum Mechanics, we reformulate Hilbert's tenth problem in the domain of integer arithmetics into either a problem involving a set of infinitely coupled differential equations or a problem involving a Shrödinger propagator with some appropriate kernel. Either way, Mathematics and Physics could be combined for Hilbert's tenth problem and for… (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)