Topological Quantum Computation

TL;DR

This paper explores topological quantum computation, which uses anyonic systems modeled by modular functors, offering a potentially more robust approach to quantum error correction compared to traditional qubit models.

Contribution

It links topological quantum field theory with quantum computation, highlighting the advantages of anyonic systems for error-resistant quantum computing.

Findings

Anyons modeled by modular functors enable topological quantum computation.

Topological error correction scales exponentially with system size.

Potential for physically realizable, error-resistant quantum computers.

Abstract

The theory of quantum computation can be constructed from the abstract study of anyonic systems. In mathematical terms, these are unitary topological modular functors. They underlie the Jones polynomial and arise in Witten-Chern-Simons theory. The braiding and fusion of anyonic excitations in quantum Hall electron liquids and 2D-magnets are modeled by modular functors, opening a new possibility for the realization of quantum computers. The chief advantage of anyonic computation would be physical error correction: An error rate scaling like $e^{-\a\l}$, where $\l$ is a length scale, and $\alpha$ is some positive constant. In contrast, the $\q$presumptive" qubit-model of quantum computation, which repairs errors combinatorically, requires a fantastically low initial error rate (about $10^{-4}$) before computation can be stabilized.