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- Tien D. Kieu
- ArXiv
- 2001

We explore in the framework of Quantum Computation the notion of Computability, 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)

- 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)

- 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)

- Tien D. Kieu
- ArXiv
- 2002

We explore in the framework of Quantum Computation the notion of computability, 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. Generalised quantum algorithms are also… (More)

- Tien D. Kieu
- Theor. Comput. Sci.
- 2004

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
- Physical review letters
- 2004

With a class of quantum heat engines which consists of two-energy-eigenstate systems undergoing, respectively, quantum adiabatic processes and energy exchanges with heat baths at different stages of a cycle, we are able to clarify some important aspects of the second law of thermodynamics. The quantum heat engines also offer a practical way, as an… (More)

- TIEN D. KIEU
- 2005

Abstract. We study a set of truncated matrices, given by Smith [8], in connection to an identification criterion for the ground state in our proposed quantum adiabatic algorithm for Hilbert’s tenth problem. We identify the origin of the trouble for this truncated example and show that for a suitable choice of some parameter it can always be removed. We also… (More)

- Tien D. Kieu
- 2004

Hilbert’s tenth problem 1 asks for a single procedure/algorithm to systematically determine if any given Diophantine equation has some positive integer solution or not. This problem has now been proved to be Turing noncomputable, indirectly through an equivalence mapping to the noncomputable Turing halting problem . Nevertheless, we have proposed and argued… (More)

- Tien D. Kieu
- ArXiv
- 2003

We employ quantum mechanical principles in the computability exploration of the class of classically noncomputable 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)

- Toby Ord, Tien D. Kieu
- Fundam. Inform.
- 2003

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