  Understanding topological matter is an outstanding challenge across several
disciplines of physical science. Programmable quantum simulators have emerged
as a powerful approach to studying such systems. While quantum spin liquids of
paradigmatic toric code type have recently been realized in the laboratory,
controlled exploration of topological phases with non-abelian excitations
remains an open problem. We introduce and analyze a new approach to simulating
topological matter based on periodic driving. Specifically, we describe a model
for a so-called Floquet spin liquid, obtained through a periodic sequence of
parallel quantum gate operations that effectively simulates the Hamiltonian of
the non-abelian spin liquid in Kitaev's honeycomb model. We show that this
approach, including the toolbox for preparation, control, and readout of
topological states, can be efficiently implemented in state-of-the-art
experimental platforms. One specific implementation scheme is based on Rydberg
atom arrays and utilizes recently demonstrated coherent qubit transport
combined with controlled-phase gate operations. We describe methods for probing
the non-abelian excitations, and the associated Majorana zero modes, and
simulate possible fusion and braiding experiments. Our analysis demonstrates
the potential of programmable quantum simulators for exploring topological
phases of matter. Extensions including simulation of Kitaev materials and
lattice gauge theories are also discussed.
