Large $k$ topological quantum computer

TL;DR

This paper investigates the universality of Chern-Simons topological quantum computers, demonstrating that for sufficiently large level k, such systems can perform universal quantum computation depending on parameters N and r.

Contribution

It proves the universality of Chern-Simons topological quantum computers for large k, clarifying how the minimum k depends on other theory parameters.

Findings

Universality is achieved for sufficiently large k.

Minimum k depends on parameters N and r.

Provides conditions for quantum computational universality.

Abstract

Chern-Simons topological quantum computer is a device that can be effectively described by the Chern-Simons topological quantum field theory and used for quantum computations. Quantum qudit gates of this quantum computer are represented by sequences of quantum $\mathcal{R}$-matrices. Its dimension and explicit form depend on the parameters of the Chern-Simons theory -- level $k$, gauge group $SU(N)$, and representation, which is chosen to be symmetric representation $[r]$. In this paper, we examine the universality of such a quantum computer. We prove that for sufficiently large $k$ it is universal, and the minimum allowed value of $k$ depends on the remaining parameters $r$ and $N$.