Freiman's Theorem describes the structure of sets of integers which are almost closed under addition. While his theorem was a major breakthrough, examples show that his description is inefficient. The Polynomial Freiman-Ruzsa Conjecture attempts to fix the inefficiencies in Freiman's Theorem. I will discuss the analogue of the Polynomial Freiman-Ruzsa Conjecture in $\mathbb{F}_{2}^{n}$, the n-dimensional vector space over the field of order 2, along with the best partial result on this conjecture, which heavily uses the $\mathbb{F}_{2}^{n}$ Fourier transform.