Hypercomputation with quantum adiabatic processes

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 are usually based on Cantor's diagonal technique used for proving non-computability of the Turing halting problem, which is related to Hilbert's tenth problem. Physical arguments are related to the limited computability of a class of quantum computation based on qubits and dimensionally finite quantum logical gates.