Gowers $U^3$ inverse theorem for quantum states

We prove a Gowers $U^3$ inverse theorem for functions $f : \mathbb{F}^n_p \rightarrow \mathbb{C}$ with $\parallel{f}\parallel_2 = 1$. This theorem tells us that if $f$ has Gowers $U^3$-norm at least $\epsilon$, then there exists a function $s$ with $\parallel s \parallel_2 = 1$ and $\parallel s \parallel_{U^3} = 1$ that has $|\langle s, f \rangle| ≥ C_{2}\epsilon^{C_1}$ for some constants $C_1$, $C_2$, depending only on $p$. A function $s$ with these properties is known in quantum information theory as a stabilizer state [1, 2]. The polynomial dependence on $\epsilon$ is facilitated by the recent breakthrough result of Green, Gowers, Marton and Tao [3],[4]. This theorem implies a polynomial time tolerant stabilizer testing algorithm for quantum states [5]. For our proof, we prove a finite field analog of the Stone-von Neumann Theorem. As far as we are aware, no proof of this theorem exists in the literature in the case that $p = 2$.