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

Quantum Physics