Is the most random pattern random? Maximizing localization in a two-dimensional lattice with engineered disorder

We investigate localization in two models: a single particle in a two-dimensional square lattice described by the tight binding Hamiltonian, and a two-dimensional square qubit lattice. It is well-known that Anderson localization occurs under suitable conditions in which the system parameters are chosen randomly from some statistical distribution. We propose a situation in which the parameters, specifically the on-site energies, are carefully chosen in such a way that a localization-quantifying parameter is maximized. We demonstrate the optimization procedure with numerical calculations in which the engineered localization significantly exceeds the average localization caused by a random distribution of the on-site energies. We explore the relation between spatial patterns and localization efficiency. Furthermore, we use perturbation theory to gain insight into the localization mechanism and obtain an improved cost function for optimization calculations, leading to enhanced localization in both the single-particle and full Hilbert spaces. Although large-scale simulations for qubit lattices are computationally infeasible, we use small-system simulations to demonstrate that results obtained using the single-particle tight binding model can be adapted to identify optimal settings for qubit lattice systems to achieve maximum decoupling between the qubits, which can be valuable for optimizing the idle-state settings on a quantum processor.