Error Correction for Non-Abelian Topological Quantum Computation

Abstract

The possibility of quantum computation using non-Abelian anyons has been considered for over a decade. However, the question of how to obtain and process information about what errors have occurred in order to negate their effects has not yet been considered. This is in stark contrast with quantum computation proposals for Abelian anyons, for which decoding algorithms have been tailor-made for many topological error-correcting codes and error models. Here, we address this issue by considering the properties of non-Abelian error correction, in general. We also choose a specific anyon model and error model to probe the problem in more detail. The anyon model is the charge submodel of D(S3) . This shares many properties with important models such as the Fibonacci anyons, making our method more generally applicable. The error model is a straightforward generalization of those used in the case of Abelian anyons for initial benchmarking of error correction methods. It is found that error correction is possible under a threshold value of 7% for the total probability of an error on each physical spin. This is remarkably comparable with the thresholds for Abelian models.

Alternate abstract:

Plain Language Summary

Non-Abelian anyons are exotic particlelike excitations in certain physical systems. They are neither bosons nor fermions but follow their own “social rules” (particle statistics). One of their promises is in topological quantum computation, where they are proposed to be used as the basis for qubits. Quantum computers built with such qubits have been believed to be naturally fault tolerant, i.e., robust against noise and having no need for “quantum error correction.” But, no concrete theoretical analyses of this assumed property have been done. In this theoretical paper, we present the first analysis of this assumption based on a toy model: Not only do we show that active error correction is indeed necessary, but we also design a method for performing such active error correction.

Anyon-based quantum computation is performed by moving anyons around each other in a manner that resembles a knitting pattern. We have shown, however, that noise will put anyons where they should not be and cause their paths to cross when they should not. Active error correction is therefore constantly required to find out where the anyons are, determine the errors that put them there, and undo those errors without disturbing the computation. Determining the errors that have occurred requires not only knowledge of the positions of the affected anyons but also whether a given pair of anyons, when brought together, will annihilate or instead fuse into a single anyon. Based on a spin-lattice model that hosts non-Abelian anyons, we have proposed an adaptive error-correction procedure that gains such information based on all information previously extracted.

As topological quantum computation based on non-Abelian anyons is being actively pursued, we believe our work is highly relevant and lays the foundation for further work on error correction in this quantum-computing architecture.