# Free fermion representation of the topological surface code

## TL;DR

This paper constructs explicit local unitary transformations that map the surface code and toric code to free fermion modes, revealing how their anyonic statistics are encoded in fermionic representations.

## Contribution

It provides a detailed construction of local unitaries mapping surface and toric codes to free fermions, clarifying their fermionic structure and anyonic properties.

## Key findings

- Surface code maps to free fermion modes
- Toric code requires additional fermionic symmetry operators
- Anyons are encoded in fermionic representations

## Abstract

The toric code is known to be equivalent to free fermions. This paper presents explicit local unitary transformations that map the $\mathbb{Z}_2$ toric and surface code --- the open boundary equivalent of the toric code --- to fermions. Through this construction it is shown that the surface code can be mapped to a set of free fermion modes, while the toric code requires additional fermionic symmetry operators. Finally, it is demonstrated how the anyonic statistics of these codes are encoded in the fermionic representations.