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In this answer: https://quantumcomputing.stackexchange.com/a/5896/19951 to a question asking how to prove (or disprove) universality, it is stated that a universal gate set "has to be capable of producing any state starting from the all zeros state."

My question: is demonstrating that a gate set is capable of producing any state starting from the zero state a sufficient condition to prove that the gate set is universal? If so, how can you guarantee that this condition ensures that any unitary can be constructed from the gate set?

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    $\begingroup$ There are several different ways that one might define universality, that are all approximately similar, but different in the details. One specification might be "the ability to implement any computation", but you conventionally start a computation from the all-zeros state... $\endgroup$
    – DaftWullie
    Commented Mar 2, 2022 at 8:06

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