I was reading McGreevy's Lecture notes Where do QFTs come from? , and on chapter 5 he talks about a duality between the $2+1d$ transverse-field Ising model (TFIM) and the $\mathbb{Z}_2$ gauge theory, where the two Hamiltonian are identified \begin{align} H_{Ising}=-\sum_{<ij>}Z_iZ_j-g\sum_i X_i \hspace{2mm}\leftrightarrow \hspace{2mm} H_{\mathbb{Z_2}}=-\sum_l\tau^x_l-g\sum_p \prod_{l\in p}\tau^z_l \end{align} through the change of variables \begin{align} Z_iZ_j\leftrightarrow \tau^x_l, \hspace{2mm} X_i\leftrightarrow \prod_{l\in p}\tau^z_l. \end{align}
However, I was wondering if there exists an operator which implements this duality in the same fashion as the Kramer-Wannier duality is implemented in the case $1+1 d$ TFIM. By that I mean that the operator \begin{align} U_{KM}\equiv\prod_{i=1}^{N-1}\left(\frac{1+iX_i}{\sqrt{2}}\right)\left(\frac{1+iZ_iZ_{i+1}}{\sqrt{2}}\right)\left(\frac{1+iX_N}{\sqrt{2}}\right) \end{align} provides the following maps for $i\neq N$ \begin{align} U_{KM} (X_i) U^{-1}_{KM}&=Z_iZ_{i+1} \\ U_{KM} (Z_iZ_{i+1})U^{-1}_{KM}&=X_i. \end{align}
Does anyone know if such operator does even exists?
