Abstract
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.
Highlights
The last few decades in condensed matter physics, starting with the discovery of the quantum Hall effect, have yielded remarkable progress in the understanding of the possible quantum states of matter
By folding manifolds in various ways, quantum origami, we demonstrate how modular transformations can be physically implemented in a single shot, i.e., with constant time overhead, and in a planar geometry with fully local interactions
The modular S, T and their combination with Rα,β implement particular types of transversal logical gates depending on particular topological states, with examples given in Table II and more detailed illustration in Appendix G
Summary
The last few decades in condensed matter physics, starting with the discovery of the quantum Hall effect, have yielded remarkable progress in the understanding of the possible quantum states of matter. Previous schemes for implementing modular transformations in physical systems involve intrusive techniques such as adiabatically varying the geometry or interactions [6,16,17,18], braiding twist defects [19], topological charge measurements [6,20], or via global rotations of a torus [11,12,21] All of these methods require a time overhead that scales polynomially with system size; that is, the time scale to carry out these operations is infinite in the thermodynamic limit. Another way to create a space that is topologically equivalent to a high genus surface is to consider a bilayer topological state in the presence of branch cuts that connect the two layers The endpoints of these branch cuts are twist defects, referred to as genons, which effectively increase the genus of the space by introducing noncontractible loops that intersect only once [see Figs. The T matrix encodes the topological twist
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