Low Overhead Universality Using Z Gates in a Uniform Constant X Field on a 1D Chain

We show that the method of quantum computation defined by applying Z-diagonal Hamiltonians in the presence of a uniform and constant external X field (as motivated, e.g., by quantum annealing using flux qubits) achieves universal quantum computation. Universality is demonstrated by construction of a universal gate set with O(1) depth overhead, and holds even if the Hamiltonian is restricted to nearest neighbor interactions on a 1D chain. 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 potential method of demonstrating quantum supremacy. Our model can achieve quantum supremacy in O(n) depth, equivalent to the circuit model despite a higher degree of homogeneity.