# Implementing the sine transform of fermionic modes as a tensor network

## TL;DR

This paper develops a tensor network implementation of the discrete sine transform of the first kind (DST-I) for fermionic modes, enabling efficient lattice computations with complex boundary conditions.

## Contribution

It introduces a recursive, algebraic, and diagrammatic method to second-quantize the DST-I as a tensor network, extending Ferris' spectral tensor network to non-trivial boundary conditions.

## Key findings

- Network complexity scales as 1.25 n log n without swap gates
- Provides a systematic approach for generalizing spectral tensor networks
- Enables efficient fermionic lattice computations with complex boundaries

## 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.