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We investigate the possibility of a many-body mobility edge in the generalized Aubry-André (GAA) model with interactions using the Shift-Invert Matrix Product States (SIMPS) algorithm [Phys. Rev. Lett. 118, 017201 (2017)]. The noninteracting GAA model is a one-dimensional quasiperiodic model with a self-duality-induced mobility edge. To search for a many-body mobility edge in the interacting case, we exploit the advantages of SIMPS that it targets many-body states in an energy-resolved fashion and does not require all many-body states to be localized for some to converge. Our analysis indicates that the targeted states in the presence of the single-particle mobility edge match neither “MBL-like” (where MBL denotes many-body localization) fully converged localized states nor the fully delocalized case in which SIMPS fails to converge. We benchmark the algorithm's output both for parameters that give fully converged, “MBL-like” localized states and for delocalized parameters where SIMPS fails to converge. In the intermediate cases, where the parameters produce a single-particle mobility edge, we find many-body states that develop entropy oscillations as a function of cut position at larger bond dimensions. These oscillations at larger bond dimensions, which are also found in the fully localized benchmark but not the fully delocalized benchmark, occur both at the band edge and center and may indicate convergence to a nonthermal state (either localized or critical). 

Given a loop or more generally 1-cycle r r of size L on a closed two-dimensional manifold or surface, represented by a triangulated mesh, a question in computational topology asks whether or not it is homologous to zero. We frame and tackle this problem in the quantum setting. Given an oracle that one can use to query the inclusion of edges on a closed curve, we design a quantum algorithm for such a homology detection with a constant running time, with respect to the size or the number of edges on the loop r r , requiring only a single usage of oracle. In contrast, classical algorithm requires \Omega(L) Ω ( L ) oracle usage, followed by a linear time processing and can be improved to logarithmic by using a parallel algorithm. Our quantum algorithm can be extended to check whether two closed loops belong to the same homology class. Furthermore, it can be applied to a specific problem in the homotopy detection, namely, checking whether two curves are not homotopically equivalent on a closed two-dimensional manifold.