  A universal family of Hamiltonians can be used to simulate any local
Hamiltonian by encoding its full spectrum as the low-energy subspace of a
Hamiltonian from the family. Many spin-lattice model Hamiltonians -- such as
Heisenberg or XY interaction on the 2D square lattice -- are known to be
universal. However, the known encodings can be very inefficient, requiring
interaction energy that scales exponentially with system size if the original
Hamiltonian has higher-dimensional, long-range, or even all-to-all
interactions. In this work, we provide an efficient construction by which these
universal families are in fact "strongly" universal. This means that the
required interaction energy and all other resources in the 2D simulator scale
polynomially in the size of the target Hamiltonian and precision parameters,
regardless of the target's connectivity. This exponential improvement over
previous constructions is achieved by combining the tools of quantum phase
estimation algorithm and circuit-to-Hamiltonian transformation in a
non-perturbative way that only incurs polynomial overhead. The simulator
Hamiltonian also possess certain translation-invariance. Furthermore, we show
that even 1D Hamiltonians with nearest-neighbor interaction of 8-dimensional
particles on a line are strongly universal Hamiltonian simulators, although
without any translation-invariance. Our results establish that analog quantum
simulations of general systems can be made efficient, greatly increasing their
potential as applications for near-future quantum technologies.
