  We show that the generating and transfer of entanglement during adiabatic
interactions can be traced to a succession of avoided energy level crossings at
which eigenvalues swap their eigenvectors. These swaps of eigenvectors weave
the entanglement in multi-partite systems. The efficiency of this weaving of
entanglement depends on the narrowness of the avoided level crossings and it
is, therefore, constraining the speed of adiabatic evolution. This relates an
adiabatic quantum computation's complexity, as measured by its minimum
duration, to that quantum computation's usage of the resource of entanglement.
To derive these results, we employ the spectral theorem to decompose
interaction Hamiltonians into rank-one projectors and we then use the adiabatic
theorem to successively adiabatically introduce these projectors. For each
projector activation, using recent exact results on the behaviour of
eigensystems under the addition of Hamiltonians, we can then non-perturbatively
calculate the entanglement dynamics. Our findings provide new tools for the
analysis and control of the dynamics of entanglement.
