  In topology, a torus remains invariant under certain non-trivial
transformations known as modular transformations. In the context of
topologically ordered quantum states of matter, these transformations encode
the braiding statistics and fusion rules of emergent anyonic excitations and
thus serve as a diagnostic of topological order. Moreover, modular
transformations of higher genus surfaces, e.g. a torus with multiple handles,
can enhance the computational power of a topological state, in many cases
providing a universal fault-tolerant set of gates for quantum computation.
However, due to the intrusive nature of modular transformations, which
abstractly involve global operations and manifold surgery, physical
implementations of them in local systems have remained elusive. Here, we show
that by folding manifolds, modular transformations can be applied in a single
shot by independent local unitaries, providing a novel class of transversal
logic gates for fault-tolerant quantum computation. Specifically, we
demonstrate that multi-layer topological states with appropriate boundary
conditions and twist defects allow modular transformations to be effectively
implemented by a finite sequence of local SWAP gates between the layers. We
further provide methods to directly measure the modular matrices, and thus the
fractional statistics of anyonic excitations, providing a novel way to directly
measure topological order.
