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We consider a model of quantum computation we call "Varying-$Z$" (V$Z$),defined by applying controllable $Z$-diagonal Hamiltonians in the presence of auniform and constant external $X$-field, and prove that it is universal, evenin 1D. Universality is demonstrated by construction of a universal gate setwith $O(1)$ depth overhead. We then use this construction to describe a circuitwhose output distribution cannot be classically simulated unless the polynomialhierarchy collapses, with the goal of providing a low-resource method ofdemonstrating quantum supremacy. The V$Z$ model can achieve quantum supremacyin $O(n)$ depth, equivalent to the random circuit sampling models despite ahigher degree of homogeneity: it requires no individually addressed$X$-control.

Low overhead universality and quantum supremacy using only $Z$-control