  The quantum spin $1/2$ XXZ chain with anisotropy parameter $\Delta=-1/2$
possesses a dynamic supersymmetry on the lattice. This supersymmetry and a
generalisation to higher spin are investigated in the case of open spin chains.
A family of non-diagonal boundary interactions that are compatible with the
lattice supersymmetry and depend on several parameters is constructed. The
cohomology of the corresponding supercharges is explicitly computed as a
function of the parameters and the length of the chain. For certain specific
values of the parameters, this cohomology is shown to be non-trivial. This
implies that the spin-chain ground states are supersymmetry singlets. Special
scalar products involving an arbitrary number of these supersymmetry singlets
for chains of different lengths are exactly computed. As a physical
application, the logarithmic bipartite fidelity of the open quantum spin $1/2$
XXZ chain with $\Delta=-1/2$ and special diagonal boundary interactions is
determined.
