Dyck Paths and Topological Quantum Computation

• URL: http://arxiv.org/abs/2306.16062v1
• Date: Wed, 28 Jun 2023 09:52:08 GMT
• Title: Dyck Paths and Topological Quantum Computation
• Authors: Vivek Kumar Singh, Akash Sinha, Pramod Padmanabhan, Indrajit Jana
• Abstract summary: We show a mapping between the fusion basis of three Fibonacci anyons, $|1rangle, |taurangle$, and the two length 4 Dyck paths. We also show braidwords in this rotated space that efficiently enable the execution of any desired single-qubit operation.
• Score: 1.3958149444453791
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• 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($\sim 10^{-3}$).

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