Fig. 1: Constraint-based implementation of \({{\mathbb{Z}}}_{2}\) mLGT with qubits.
The \({{\mathbb{Z}}}_{2}\) gauge structure emerges from the dominant local-pseudogenerator (LPG) interaction on the honeycomb lattice introduced in (a). A vertex contains matter \({\hat{a}}_{{{{{{{{\boldsymbol{j}}}}}}}}}\) qubits (blue) and shares link \({\hat{\tau }}_{\langle {{{{{{{\boldsymbol{i}}}}}}}},{{{{{{{\boldsymbol{j}}}}}}}}\rangle }^{x}\) qubits (red) with neighboring vertices. All qubits connected to a vertex interact pairwise with strength 2V. In a Rydberg atom array experiment the qubits are implemented by individual atoms in optical tweezers, which are assigned the role of matter or link depending on the position in the lattice. Here, the ground- and Rydberg state of the atoms, \(\left\vert g\right\rangle\) and \(\left\vert r\right\rangle\), encode qubit states, which are coupled by an off-resonant drive Ω to induce effective interactions. To realize equal strength nearest neighbor, two-body Rydberg–Rydberg interactions, the matter atoms can be elevated out of plane. In (b) we introduce the notation for the \({{\mathbb{Z}}}_{2}\) mLGT, for which the Hilbert space constraint is given by Gauss’s law \({\hat{G}}_{{{{{{{{\boldsymbol{j}}}}}}}}}=+1\). We illustrate the electric field \({\tau }_{\langle {{{{{{{\boldsymbol{i}}}}}}}},{{{{{{{\boldsymbol{j}}}}}}}}\rangle }^{x}=+1\) (\({\tau }_{\langle {{{{{{{\boldsymbol{i}}}}}}}},{{{{{{{\boldsymbol{j}}}}}}}}\rangle }^{x}=-1\)) with flat (wavy) red lines and the matter site occupation nj = 0 (nj = 1) with empty (full) blue dots. c shows the notation for the QDM subspace with exactly one dimer per vertex. d illustrates how the distinct subspaces are energetically separated by the LPG term \(V{\hat{W}}_{{{{{{{{\boldsymbol{j}}}}}}}}}\). The two quantum dimer subspaces are disconnected when the matter is static, which can be exactly realized by the absence of matter atoms in (a) and setting \((2{\hat{a}}_{{{{{{{{\boldsymbol{j}}}}}}}}}^{{{{\dagger}}} }{\hat{a}}_{{{{{{{{\boldsymbol{j}}}}}}}}}-1)=\pm \!1\) in \(V{\hat{W}}_{{{{{{{{\boldsymbol{j}}}}}}}}}\).
