Exploiting geometric degrees of freedom in topological quantum computing

In a topological quantum computer, braids of non-Abelian anyons in a $\left(2+1\right)$-dimensional space time form quantum gates, whose fault tolerance relies on the topological, rather than geometric, properties of the braids. Here we propose to create and exploit redundant geometric degrees of freedom to improve the theoretical accuracy of topological single- and two-qubit quantum gates. We demonstrate the power of the idea using explicit constructions in the Fibonacci model. We compare its efficiency with that of the Solovay-Kitaev algorithm and explain its connection to the leakage errors reduction in an earlier construction [H. Xu and X. Wan, Phys. Rev. A 78, 042325 (2008)].

Figure 1

(Color online) (a) Mapping two qubits ($a_1\text{−}{a}_4$ and $a_5\text{−}{a}_8$) into one qubit consisting of four composite anyons $A_1\text{−}{A}_4$. (b) Braid $P_2$ of the composite anyons $A_1$ to $A_4$. As explained in the text, we only move the composite anyon $A_2$ to braid with the composite anyons $A_3$ and $A_4$ and return $A_2$ back to the original position at the end of the braid.

Figure 2

(Color online) A scheme to realize a generic controlled-rotation gate in the Fibonacci anyon model. We consider two qubits composed of $a_1\text{−}{a}_8$. $P_2$, which realizes a two-qubit controlled-phase gate, is a braid acting on the effective single qubit formed by composite anyons $A_1\text{−}{A}_4$. $G_3$ and $P_3$ are braids of single-qubit gates that modify the controlled-phase gate to a generic controlled-rotation gate. Note that we choose the convention of the time direction from left to right.