Abstract
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.
Highlights
After the integer quantum Hall effect had been theoretically understood, the richness of the unexpectedly discovered fractional quantum Hall effect has left no doubt that the addition of strong interactions can lead to even more remarkable topological phenomena [1,2,3,4]
We propose a building block for Z2 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
The elementary Z2 building block we propose for superconducting qubits, see Fig. 1, resembles the building block proposed [57] and realized [58] earlier with ultracold atoms in optical lattices, see Ref. [59] and a proposal with two-species fermionic atoms [60]
Summary
After the integer quantum Hall effect had been theoretically understood, the richness of the unexpectedly discovered fractional quantum Hall effect has left no doubt that the addition of strong interactions can lead to even more remarkable topological phenomena [1,2,3,4] These include topological ground-state degeneracies and anyonic excitations with non-Abelian braiding statistics. The robust topological ground-state degeneracy on a torus can be understood as a result of nonlocal gauge excitations, which represent a nontrivial pattern of entanglement in the ground state This close connection is most clearly demonstrated in Kitaev’s toric code [29], which represents an exactly solvable Z2 lattice gauge theory (LGT) [30].
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