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In topology, a torus remains invariant under certain non-trivialtransformations known as modular transformations. In the context oftopologically ordered quantum states of matter, these transformations encodethe braiding statistics and fusion rules of emergent anyonic excitations andthus serve as a diagnostic of topological order. Moreover, modulartransformations of higher genus surfaces, e.g. a torus with multiple handles,can enhance the computational power of a topological state, in many casesproviding a universal fault-tolerant set of gates for quantum computation.However, due to the intrusive nature of modular transformations, whichabstractly involve global operations and manifold surgery, physicalimplementations of them in local systems have remained elusive. Here, we showthat by folding manifolds, modular transformations can be applied in a singleshot by independent local unitaries, providing a novel class of transversallogic gates for fault-tolerant quantum computation. Specifically, wedemonstrate that multi-layer topological states with appropriate boundaryconditions and twist defects allow modular transformations to be effectivelyimplemented by a finite sequence of local SWAP gates between the layers. Wefurther provide methods to directly measure the modular matrices, and thus thefractional statistics of anyonic excitations, providing a novel way to directlymeasure topological order.

Quantum Origami: Transversal Gates for Quantum Computation and Measurement of Topological Order