  We explore in the framework of Quantum Computation the notion of {\em
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 quantum
adiabatic evolution are employed. If this algorithm could be physically
implemented, as much as it is valid in principle--that is, if certain
hamiltonian and its ground state can be physically constructed according to the
proposal--quantum computability would surpass classical computability as
delimited by the Church-Turing thesis. It is thus argued that computability,
and with it the limits of Mathematics, ought to be determined not solely by
Mathematics itself but also by Physical Principles.
