Implementing the sine transform of fermionic modes as a tensor network

Abstract

Based on the algebraic theory of signal processing, we recursively decompose the discrete sine transform of first kind (DST-I) into small orthogonal block operations. Using a diagrammatic language, we then second-quantize this decomposition to construct a tensor network implementing the DST-I for fermionic modes on a lattice. The complexity of the resulting network is shown to scale as $\frac 54 n \log n$ (not considering swap gates), where $n$ is the number of lattice sites. Our method provides a systematic approach of generalizing Ferris' spectral tensor network for non-trivial boundary conditions.