  We investigate the relative volume of parent Hamiltonians having a target
ground state up to some fixed error $\epsilon$, a quantity which sets a
benchmark on the performance of quantum simulators. For vanishing error, this
relative volume is of measure zero, whereas for a generic $\epsilon$ we show
that it increases with the dimension of the Hilbert space. We also address the
volume of parent Hamiltonians when they are restricted to be local. For
translationally invariant Hamiltonians, we provide an upper bound to their
relative volume. Finally, we estimate numerically the relative volume of parent
Hamiltonians when the target state is the ground state of the Ising chain in a
transverse field.
