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 $[-\pi, \pi]$.


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