  Kitaev's toric code is an exactly solvable model with
$\mathbb{Z}_2$-topological order, which has potential applications in quantum
computation and error correction. However, a direct experimental realization
remains an open challenge. Here, we propose a building block for $\mathbb{Z}_2$
lattice gauge theories coupled to dynamical matter and demonstrate how it
allows for an implementation of the toric-code ground state and its topological
excitations. This is achieved by introducing separate matter excitations on
individual plaquettes, whose motion induce the required plaquette terms. The
proposed building block is realized in the second-order coupling regime and is
well suited for implementations with superconducting qubits. Furthermore, we
propose a pathway to prepare topologically non-trivial initial states during
which a large gap on the order of the underlying coupling strength is present.
This is verified by both analytical arguments and numerical studies. Moreover,
we outline experimental signatures of the ground-state wavefunction and
introduce a minimal braiding protocol. Detecting a $\pi$-phase shift between
Ramsey fringes in this protocol reveals the anyonic excitations of the
toric-code Hamiltonian in a system with only three triangular plaquettes. Our
work paves the way for realizing non-Abelian anyons in analog quantum
simulators.
