Universal Hamiltonian simulators in one and two dimensions

encodes the physics of a target Hamiltonian H. Some families of 2D spin-lattice Hamiltonians-such as the Heisenberg or XY model on the square lattice-are known to be "universal" simulators, in the sense that they can simulate any target Hamiltonian. Unfortunately, while the known simulations of 1D or 2D target Hamiltonians are efficient, they require resources growing exponentially with system sizes for target Hamiltonians with general connectivity. This leaves many important cases such as 3D or all-to-all-connected target Hamiltonians out of reach in practice. Here, we remedy this situation and show how the known 2D universal families of Hamiltonians and a new 1D family can simulate all target local Hamiltonians, with only polynomial-scaling overhead. This exponential improvement is achieved by a nonperturbative method combining the quantum phase-estimation algorithm and the circuit-to-Hamiltonian construction. Furthermore, all Hamiltonians efficiently simulable by quantum circuits, including nonlocal ones, also have efficient analog simulators in our 2D and 1D universal families. Our work establishes that analog simulations of general Hamiltonians can be made efficient, significantly expanding the application potential of analog Hamiltonian simulations in near-term quantum technologies.