Unfolding the Toric Code Model with Emergent Qubits

We present the idea of emergent qubits by an exact model construction on a trestle, also generalized to arbitrary graphs. The corresponding eigenstates are quantum paramagnetic, with free multipolar moments. We rigorously transform the toric code model on a torus, cylinder and sheet into emergent qubits, writing all the eigenstates exactly. We devise exact quantum circuits for the toric code and other eigenstates described here. The depth of the circuit for toric code eigenstates on torus grows linearly with the total number of qubits, as compared to the sublinear growth on cylinder or sheet.