For the case of Hamiltonians, rather than gates the answer is trivially yes: you simply enumerate the independent elements of the Lie algebra. Since the Lie algebra is a vector space with the addition of the Lie bracket operator. Since the space is finite, it has a finite basis, and which can easily be checked as to whether it is closed or open under the Lie bracket operation. Simply checking the Lie bracket of all pairs of orthogonal operators can be done in time polynomial in the dimensionality of the space, and a suitable operator basis can be found by the Gram-Schmidt method.

For gates, you don't really have the same option to resort to infinitesimals straight off, and need to construct gates with irrational eigenvalues so that you can arbitrarily well approximate the required infinitesimal generators. I guess that there is a relatively simple way to do this, but it is not immediately obvious to me.

In any case, taking the log of the gates to obtain a set of operators which generate them when exponentiated and checking whether these generated the full Lie algebra would provide a simple necessary but not sufficient criteria for universality.

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