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

TL;DR

This paper introduces a method to implement modular transformations in topological quantum states using local unitaries, enabling fault-tolerant quantum gates and direct measurement of topological order.

Contribution

It presents a novel approach to realize modular transformations via local SWAP gates in layered topological states, facilitating fault-tolerant quantum computation and measurement of topological order.

Findings

Modular transformations can be implemented with local unitaries in layered systems.

The method enables direct measurement of topological order and anyonic statistics.

Provides a new pathway for fault-tolerant quantum computation using topological states.

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.