Tien D. Kieu

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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)
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 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)
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)
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)
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)
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)
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)