# A tree tensor network approach to simulating Shor's algorithm

## TL;DR

This paper introduces a novel tensor network method to simulate and analyze the entanglement structure of Shor's algorithm, revealing how entanglement scales with system size and modular periodicity.

## Contribution

It presents a tree tensor network approach that efficiently maps to a matrix product state, enabling detailed study of entanglement in Shor's wavefunction.

## Key findings

- Entanglement grows exponentially with qubits before saturation.
- Saturation occurs at a scale proportional to modular periodicity.
- The method captures non-local entanglement structure explicitly.

## Abstract

Simulating quantum systems constructively furthers our understanding of qualitative and quantitative features which may be analytically intractable. In this letter, we directly simulate and explore the entanglement structure present in a paradigmatic example of quantum information: Shor's wavefunction. The methodology employed is a dynamical tensor network which is initially constructed as a tree tensor network, inspired by the modular exponentiation quantum circuit, and later efficiently mapped to a matrix product state. Utilizing the Schmidt number as a local entanglement metric, our construction explicitly captures the wavefunction's non-local entanglement structure and an entanglement scaling relation is discovered. Specifically, we see that entanglement across a bipartition grows exponentially in the number of qubits before saturating at a critical scale which is proportional to the modular periodicity.