Everything is a quantum Ising model

This work shows that any $k$-local Hamiltonian of qubits can be obtained from a 4-state 'Ising' model with $k$-local diagonal interactions and a single-site transverse field -- giving a new theoretical and experimental handle on quantum matter. In particular, the classical Ising interactions can be determined by replacing each Pauli operator with a $4 \times 4$ diagonal matrix. Subsequently tuning a large transverse field projects out two of the four states, recovering the original qubit model, with qudit generalizations. This leads to striking correspondences, such as the spin-1/2 XY and Heisenberg models arising from the large-field limit of 3-state and 4-state Potts models, respectively. Similarly, the Kitaev honeycomb model emerges from classical interactions which enforce loop states on the honeycomb lattice. These generalized Ising models also display rich physics for smaller fields, including quantum criticality and topological phases of matter. This work expands what is experimentally achievable by showing how to realize any quantum spin model using only diagonal interactions and a tuneable field -- ingredients found in, e.g., tweezer arrays of Rydberg atoms or polar molecules. More broadly, 4-state spins can also be encoded in the positions of itinerant particles, exemplified by a Bose-Hubbard model realizing the Kitaev honeycomb model -- giving an experimental path to its $\mathbb Z_2$ and non-Abelian topological quantum liquids.