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Universal resources for approximate and stochastic measurement-based quantum computation

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We investigate which families of quantum states can be used as resources for approximate and/or stochastic universal measurement-based quantum computation, in the sense that single-qubit operations and classical communication are sufficient to prepare (with some fixed precision and/or probability) any quantum state from the initial resource. We find entanglement-based criteria for non-universality in the approximate and/or stochastic case. By applying them, we are able to discard some families of states as not universal also in this weaker sense. Finally, we show that any family $Sigma$ of states that is \'close\' to an (approximate and/or stochastic) universal family $Gamma$ is approximate and stochastic universal, and we prove that if $Gamma$ was efficiently universal then also $Sigma$ is.