# Reconstructing classical data from quantum feature maps

In the paper [Supervised learning with quantum enhanced feature spaces (Nature, arxiv) by Havlicek et al., a feature map is defined by $$| \Phi(\bar{x})\rangle=U_{\Phi(\bar x)}H^{\otimes n} U_{\Phi(\bar x)}H^{\otimes n} |0\rangle^{\otimes n},$$ where $$U_{\Phi(\bar x)}=\exp(i \sum_{S\subseteq [n]} \phi_S(\bar x) \prod_{i\in S} Z_i)$$.

I'm interested in the case where $$|S| \leq 2$$, $$\phi_{\{i\}}(\bar x)=x_i$$ and $$\phi_{\{i,j\}}(\bar x)=(\pi-x_i)(\pi-x_j)$$.

My question: given $$|\Phi( \bar x)\rangle$$, is there any way to reconstruct $$\bar x$$? You may assume that each $$x_i$$ is from $$[-\pi, \pi]$$.