An alternating-minimization method for preparing low-energy states

Preparing low energy states is a central challenge in quantum computing and quantum complexity theory. Several known approaches to prepare low energy states often get stuck in suboptimal states, such as high energy eigenstates (or low variance high energy states). We develop a heuristic method to go past this barrier for local Hamiltonian systems with relatively low frustration, by taking advantage of the fact that such systems come with multiple Hamiltonians that agree on the low-energy subspaces. We establish an energy-based uncertainty principle, which shows that these Hamiltonians in fact do not have common eigenstates in the high energy regime. This allows us to run energy lowering steps in an alternating manner over the Hamiltonians. We run numerical simulations to check the performance of the `alternating' algorithm on small system sizes, for the 1D AKLT model and instances of Heisenberg model on general graphs. We also formulate a version of the energy-based uncertainty principle using sparse Hamiltonians, which shows a quadratically larger variance at higher energies and hence leads to a larger energy change. We use this version to simulate the method on energy profiles with high energy barriers.