Quantum Origami: Applying Fault-tolerant Transversal Gates and Measuring Topological Order

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 and introducing twist defects, modular transformations can be reduced to independent local unitaries, specifically, a finite sequence of local SWAP gates between the layers, providing a novel class of transversal logic gates in topological codes and leading to universal gate set. We further propose methods to directly measure the modular matrices, and thus the fractional statistics of anyonic excitations, providing a novel way to directly measure topological order.