  Recent work has characterised rigorously what it means for one quantum system
to simulate another, and demonstrated the existence of universal Hamiltonians
-- simple spin lattice Hamiltonians that can replicate the entire physics of
any other quantum many body system. Previous universality results have required
proofs involving complicated `chains' of perturbative `gadgets'. In this paper,
we derive a significantly simpler and more powerful method of proving
universality of Hamiltonians, directly leveraging the ability to encode quantum
computation into ground states. This provides new insight into the origins of
universal models, and suggests a deep connection between universality and
complexity. We apply this new approach to show that there are universal models
even in translationally invariant spin chains in 1D. This gives as a corollary
a new Hamiltonian complexity result, that the local Hamiltonian problem for
translationally-invariant spin chains in one dimension with an
exponentially-small promise gap is PSPACE-complete. Finally, we use these new
universal models to construct the first known toy model of 2D--1D holographic
duality between local Hamiltonians.
