Quantum many-body systems are computationally challenging because the Hilbert space dimension grows exponentially with system size. Tensor network methods, particularly Tree Tensor Networks (TTNs), provide efficient representations by exploiting the hierarchical entanglement structure of quantum states. This thesis investigates algebraic operations on TTNs, focusing on state addition algorithms and their implementation.
We present two approaches for TTN addition: direct addition combines tensor networks through block-diagonal concatenation, while basis extension incorporates orthogonalized components from one state into another’s bond basis. The basis extension method, inspired by time-dependent variational principle techniques from Yang and White [1], uses gauge freedom and canonical forms to construct expanded representations while preserving the original state. Both algorithms are implemented in the PyTreeNet framework [2]. We analyze computational complexity, examining how bond dimensions scale with each method and the trade-offs between exactness and efficiency. This work provides foundational tools for quantum state manipulation in tree tensor network representations.