  We show how the localization landscape, originally introduced to bound low
energy eigenstates of disordered wave media and many-body quantum systems, can
form the basis for hardware-efficient quantum algorithms for solving binary
optimization problems. Many binary optimization problems can be cast as finding
low-energy eigenstates of Ising Hamiltonians. First, we apply specific
perturbations to the Ising Hamiltonian such that the low energy modes are
bounded by the localization landscape. Next, we demonstrate how a variational
method can be used to prepare and sample from the peaks of the localization
landscape. Numerical simulations of problems of up to $10$ binary variables
show that the localization landscape-based sampling can outperform QAOA
circuits of similar depth, as measured in terms of the probability of sampling
the exact ground state.
