  Quantum adiabatic algorithm is a method of solving computational problems by
evolving the ground state of a slowly varying Hamiltonian. The technique uses
evolution of the ground state of a slowly varying Hamiltonian to reach the
required output state. In some cases, such as the adiabatic versions of
Grover's search algorithm and Deutsch-Jozsa algorithm, applying the global
adiabatic evolution yields a complexity similar to their classical algorithms.
However, using the local adiabatic evolution, the algorithms given by J. Roland
and N. J. Cerf for Grover's search [ Phys. Rev. A. {\bf 65} 042308(2002)] and
by Saurya Das, Randy Kobes and Gabor Kunstatter for the Deutsch-Jozsa algorithm
[Phys. Rev. A. {\bf 65}, 062301 (2002)], yield a complexity of order $\sqrt{N}$
(where N=2$^{\rm n}$ and n is the number of qubits). In this paper we report
the experimental implementation of these local adiabatic evolution algorithms
on a two qubit quantum information processor, by Nuclear Magnetic Resonance.
