Exploiting the Hermitian symmetry in tensor network algorithms

Exploiting symmetries in tensor network algorithms plays a key role for reducing the computational and memory costs. Here we explain how to incorporate the Hermitian symmetry in double-layer tensor networks, which naturally arise in methods based on projected entangled-pair states. For real-valued tensors the Hermitian symmetry defines a Z2 symmetry on the combined bra and ket auxiliary level of the tensors. By implementing this symmetry, a speedup of the computation time by up to a factor 4 can be achieved, while expectation values of observables and reduced density matrices remain Hermitian by construction. Benchmark results based on the corner transfer matrix renormalization group and higher-order tensor renormalization group are presented. We also discuss how to implement the Hermitian symmetry in the complex case, where a similar speedup can be achieved.