The properties of out-of-equilibrium dynamics of interacting many-body quantum systems are intrinsically hard to investigate.The reason is that quantum entanglement, generated by the presence of many interacting quantum degrees of freedom, generally requires resources that grow exponentially with the system size. This prevents exact simulations of generic interacting systems for large system sizes and intermediate and long time scales on classical computers. Many open questions in many-body quantum physics remain in this domain. To overcome this barrier, we have developed various, radically different, approximate algorithms whose underlying idea is to represent quantum states with less than exponential (in system-size) degrees of freedom, keeping track only of the most relevant ones. Our project uses two kinds of algorithms.
(i) Tensor Networks. The presence of disorder can exponentially suppress long-range entanglement in many-body quantum systems, inducing a Many-Body Localized (MBL) phase where quantum states admit a compressed representation using tensor networks. We have developed two algorithms using tensor networks to study the entanglement statistics of disordered quantum spin chains in (or near) the MBL phase. The first (xDMRG) calculates excited eigenstates of spin chain models by adapting the Density Matrix Renormalization Group (DMRG) algorithm, that is normally used for ground states. The second (fLBIT) indirectly simulates the dynamics of MBL systems, similarly to the Time-Evolving Block Decimation (TEBD) algorithm. Both algorithms rely heavily on tensor contractions and singular-value decompositions.
(ii) Local-Information Algorithm. Simulating non-localized systems while keeping track of all degrees of freedom is generally not possible. We have developed a new code that systematically discards long-range correlations, which allows us to approximate the dynamics of interacting many-body quantum systems with an error that is controlled by the length scale after which we disregard entanglement. This code is based on recent insights by members of our group and their collaborators. Specifically, we aim to solve the von Neumann equation (VNE) that describes the time evolution of density matrices by using the locality of the Hamiltonians we investigate. Instead of solving the VNE for the entire system, which is a formidable task even for small system sizes, our methods decompose the whole system into smaller subsystems and solve the corresponding VNE on each subsystem in parallel. Such an algorithm is particularly well suited for parallelization via MPI. The local-information algorithm is versatile and can be applied to various physical contexts, such as localized, diffusive, super-diffusive, and ballistic hydrodynamic behaviors. Furthermore, it is suitable for both closed and open quantum systems.