Advanced Tensor Network Algorithms for Quantum Many-Body System Simulation

Tensor network methods are a family of state-of-the-art numerical methods for simulating quantum many-body systems. This thesis contributes to the advancement of tensor network techniques through three objectives: the development and open-source implementation of advanced algorithms, benchmarking and comparing different tensor network approaches, and, finally, their application to problems in quantum many-body physics. We present Quantum TEA, an open-source tensor network emulator providing an HPC-ready collection of algorithms for quantum many-body system simulation. We use Quantum TEA to perform large-scale variational ground-state search benchmarks, systematically comparing the performance of different tensor network ans¨atze. Using these insights, we explore two distinct phenomena arising in the ground states of quantum many-body Hamiltonians: first, in the quantum Ising model on a three-legged ladder we find the first instance in which topological frustration shifts the position of a quantum critical point; and second, we provide numerical evidence for the emergence of supersolid crystals in the Bose–Hubbard model with long-range hoppings, complementing the experimental observations with dipolar excitons. Moving to the mixed-state scenario, we build a toolbox capable of computing entanglement monotones and other global properties of the density matrix, otherwise difficult to access in a many-body setting. We construct the algorithms for simulating thermal equilibrium and open system dynamics within this framework, and demonstrate their capabilities by studying finite-temperature Rydberg arrays and the boundary-driven XXZ chains. Finally, we show how our thermal equilibrium algorithm can be exploited to extract the low-energy part of a many-body Hamiltonian’s spectrum, finding that this approach outperforms the variational excited-state search away from quantum criticality. Altogether, the thesis directly showcases the versatility of tensor network methods and highlights that selecting an appropriate simulation strategy requires carefully leveraging the strengths of each framework. This, in turn, underscores the essential role that flexible open-source software tools provide to the research community.