  We investigate which quantum states can serve as universal resources for
approximate and stochastic measurement-based quantum computation, in the sense
that any quantum state can be generated from a given resource by means of
single-qubit (local) operations assisted by classical communication. More
precisely, we consider the approximate and stochastic generation of states,
resulting e.g. from a restriction to finite measurement settings or from
possible imperfections in the resources or local operations. We show that
entanglement-based criteria for universality obtained for the exact,
deterministic case can be lifted to the much more general approximate,
stochastic case, moving from the idealized situation considered in previous
works, to the practically relevant context of non-perfect state preparation. We
find that any entanglement measure fulfilling some basic requirements needs to
reach its maximum value on some element of an approximate, stochastic universal
family of resource states, as the resource size grows. This allows us to rule
out various families of states as being approximate, stochastic universal. We
provide examples of resources that are efficient approximate universal, but not
exact deterministic universal. We also study the robustness of universal
resources for measurement-based quantum computation under realistic assumptions
about the (imperfect) generation and manipulation of entangled states, giving
an explicit expression for the impact that errors made in the preparation of
the resource have on the possibility to use it for universal approximate and
stochastic state preparation. Finally, we discuss the relation between our
entanglement-based criteria and recent results regarding the uselessness of
states with a high degree of geometric entanglement as universal resources.
