The localization landscape

Localization of waves due to destructive interference in disordered media - Anderson localization - has been a subject of intense investigation for more than 50 years. The original paper has now been cited more than 15,000 times according to Google Scholar. Being a property of linear, non-interacting Hamiltonians subjected to a random potential, one might think that everything there is to know about this problem would have already been studied to death a long time ago, with current work focusing on understanding peculiarities arising under special circumstances.

Delightfully, this is not the case! Back in February, this preprint caught my attention due to its keywords of many-body localization and persistent homology. Specifically, the authors have used persistent homology to characterize the shape of the "localization landscape" of a many-body Hamiltonian. 

What is the localization landscape?

The localization landscape was first proposed by Filoche and Mayboroda in an article in PNAS which, despite its broad applicability to understanding wave transport in a variety of settings (condensed matter, photonics, acoustics), did not attract as much interest as other arguably more specialized areas such as non-Hermitian systems or topological edge states. In short, the localization landscape u of a Hamiltonian $\mathcal{H}$ satisfying certain properties is the solution of the linear equation

$$\mathcal{H} u = \mathcal{1},$$

where $\mathcal{1}$ is a vector with all elements equal to 1. u corresponds to the steady-state response of the medium to a uniformly-distributed source.

Left: The localization landscape of a disordered two-dimensional wave medium. Right: Five lowest energy eigenstates of the Hamiltonian, which are localized to distinct peaks of the landscape bounded by lines of small $u$ (red). For more details, read the paper!

Remarkably, it can be shown that $u$ bounds the spatial extend of the low energy eigenmodes of $\mathcal{H}$. Thus, instead of solving the eigenvalue problem and plotting its low energy modes individually to find out where they are localized, plotting $u$ alone is enough to find all the effective low-energy valleys in a medium. Moreover, as a solution to a linear system of equations, $u$ is easier to obtain than the eigenstates themselves. A more rigorous discussion (including the properties that must be satisfied by $\mathcal{H}$ can be found in the PNAS article, which is a pleasant and accessible read.

Interest is now growing in localization landscapes, thanks to recent generalizations that can bound the spatial extent of modes residing in the middle of the energy spectrum, eigenvectors of real symmetric matrices and modes of many-body interacting quantum systems. The latter remarkably shows that useful landscape functions are not limited to low-dimensional wave media, but can also bound the spreading of wavefunctions in the high-dimensional Fock space of many-body quantum systems.

A natural question that arises from these recent works is whether the localization landscape might have some potential applications in the context of quantum computing and noisy intermediate-scale quantum (NISQ) processors. For example, finding the ground state of a generic many-body Hamiltonian is a computationally challenging problem, and methods for NISQ devices such as the variational quantum eigensolver may fail to converge due to the presence of vanishing gradients or sub-optimal local minima. 

Can the localization landscape offer an easier, more hardware-efficient way to sample from low-energy solutions of a hard-to-solve Hamiltonian? Watch this space to find out!