Monodromy analysis of the computational power of the Ising topological quantum computer

TL;DR

This paper analyzes the computational capabilities of the Ising topological quantum computer, demonstrating that braiding operations only realize a subset of Clifford gates and thus have limited computational power.

Contribution

It provides a detailed monodromy analysis showing that braiding in the Ising topological quantum computer cannot implement all Clifford gates, revealing fundamental limitations.

Findings

Braiding preserves the n-qubit Pauli group.

The image of the braid group is a proper subgroup of the Clifford group.

Not all Clifford gates can be realized by braiding in the Ising model.

Abstract

We show that all quantum gates which could be implemented by braiding of Ising anyons in the Ising topological quantum computer preserve the n-qubit Pauli group. Analyzing the structure of the Pauli group's centralizer, also known as the Clifford group, for n\geq 3 qubits, we prove that the image of the braid group is a non-trivial subgroup of the Clifford group and therefore not all Clifford gates could be implemented by braiding. We show explicitly the Clifford gates which cannot be realized by braiding estimating in this way the ultimate computational power of the Ising topological quantum computer.