Dyck Paths and Topological Quantum Computation

TL;DR

This paper establishes a novel connection between Dyck paths and Fibonacci anyons, enabling the construction of a topological quantum computing platform with demonstrated stability and efficient single-qubit operations.

Contribution

It introduces a mapping between Fibonacci anyon fusion bases and Dyck paths, facilitating new methods for topological quantum computation using spin chains.

Findings

The system is gapped and stable against noise.

Dyck paths correspond to Fibonacci fusion basis states.

Efficient braidwords enable precise single-qubit operations.

Abstract

The fusion basis of Fibonacci anyons supports unitary braid representations that can be utilized for universal quantum computation. We show a mapping between the fusion basis of three Fibonacci anyons, $\{|1\rangle, |\tau\rangle\}$, and the two length 4 Dyck paths via an isomorphism between the two dimensional braid group representations on the fusion basis and the braid group representation built on the standard $(2,2)$ Young diagrams using the Jones construction. This correspondence helps us construct the fusion basis of the Fibonacci anyons using Dyck paths as the number of standard $(N,N)$ Young tableaux is the Catalan number, $C_N$ . We then use the local Fredkin moves to construct a spin chain that contains precisely those Dyck paths that correspond to the Fibonacci fusion basis, as a degenerate set. We show that the system is gapped and examine its stability to random noise thereby establishing its usefulness as a platform for topological quantum computation. Finally, we show braidwords in this rotated space that efficiently enable the execution of any desired single-qubit operation, achieving the desired level of precision($\sim 10^{-3}$).