  Analogue Hamiltonian simulation is a promising application of quantum
computing in the NISQ era. Recent work has demonstrated the existence of
universal Hamiltonians -- simple spin lattice models that can simulate any
other quantum many body system. Until now proofs of universality have relied on
explicit constructions, which are shown to be able to replicate the physics of
some arbitrary target Hamiltonian. In this work, we derive necessary and
sufficient conditions for a family of Hamiltonians to be universal. The proof
relies on complexity theoretic concepts -- leveraging the ability to encode
quantum computation into the ground states of Hamiltonians. This offers a far
simpler route to proving universality, and provides insight into the
relationship between complexity and universality.
