  We provide several examples and an intuitive diagrammatic representation
demonstrating the use of two-qubit unitary transformations for mapping coupled
spin Hamiltonians to simpler ones and vice versa. The corresponding dualities
may be exploited to identify phase transition points or to aid the
diagonalization of such Hamiltonians. For example, our method shows that a
suitable one-parameter family of coupled Hamiltonians whose ground states
transform from an initially factorizing state to a final cluster state on a
lattice of arbitrary dimension is dual to a family of trivial decoupled
Hamiltonians containing local on-site terms only. As a consequence, the minimum
enery gap (which determines the adiabatic run-time) does not scale with system
size, which facilitates an efficient and simple adiabatic preparation of e.g.
the two-dimensional cluster state used for measurement-based quantum
computation.
