  We show that it is possible to use a classical computer to efficiently
simulate the adiabatic evolution of a quantum system in one dimension with a
constant spectral gap, starting the adiabatic evolution from a known initial
product state. The proof relies on a recently proven area law for such systems,
implying the existence of a good matrix product representation of the ground
state, combined with an appropriate algorithm to update the matrix product
state as the Hamiltonian is changed. This implies that adiabatic evolution with
such Hamiltonians is not useful for universal quantum computation. Therefore,
adiabatic algorithms which are useful for universal quantum computation either
require a spectral gap tending to zero or need to be implemented in more than
one dimension (we leave open the question of the computational power of
adiabatic simulation with a constant gap in more than one dimension).
