# Matrix product state algorithms for Gaussian fermionic states

## TL;DR

This paper adapts matrix product state algorithms to efficiently simulate large systems of non-interacting fermions, achieving exponential speedup and enabling solutions for systems with up to one million sites.

## Contribution

It introduces MPS-based methods tailored for Gaussian fermionic states, significantly improving simulation efficiency over traditional approaches.

## Key findings

- Able to simulate systems of up to one million sites
- Achieves exponential speedup in entanglement entropy scaling
- Handles an effective bond dimension of 10^15

## Abstract

While general quantum many-body systems require exponential resources to be simulated on a classical computer, systems of non-interacting fermions can be simulated exactly using polynomially scaling resources. Such systems may be of interest in their own right, but also occur as effective models in numerical methods for interacting systems, such as Hartree-Fock, density functional theory, and many others. Often it is desirable to solve systems of many thousand constituent particles, rendering these simulations computationally costly despite their polynomial scaling. We demonstrate how this scaling can be improved by adapting methods based on matrix product states, which have been enormously successful for low-dimensional interacting quantum systems, to the case of free fermions. Compared to the case of interacting systems, our methods achieve an exponential speedup in the entanglement entropy of the state. We demonstrate their use to solve systems of up to one million sites with an effective MPS bond dimension of 10^15.