Universal Quantum Gate Compilation in $SU(2)_k$ Anyon Models via Multiple-Braiding

Abstract:We investigate the capability of multiple-braiding in $SU(2)_k$ anyon models to realize universal quantum computation. The multiple elementary braiding matrices (MEBMs) are derived from the $q$-deformed representation theory of $SU(2)$. Through a general analysis of the MEBMs, we find that multiple-braiding loses its universality only when the braiding multiplicity $m$ causes the MEBMs to collapse to a scalar (up to a global phase). Our numerical analysis of the MEBMs of $SU(2)_k$ anyon models with $k = 3, 5, 6, 7$, and $m \in [2, 9]$, shows that the values of $m$ at which the braid group representation fails to be dense agree exactly with the theoretical prediction. For those $m$ that support universality, single-qubit gates are compiled to high precision by a genetic algorithm-enhanced Solovay-Kitaev algorithm (GA-enhanced SKA), and a genetic algorithm (GA) search with progressively increasing braid length yields an approximately the local equivalence class $[CNOT]$. Notably, even-order braiding operations offer a physical advantage by reducing the number of non-Abelian anyons required in braiding-based topological quantum computation (TQC). Our analytical and numerical results together provide strong evidence for identifying which braiding multiplicities $m$ support universal quantum computation in $SU(2)_k$ anyon models.

arXiv:2602.15324 [quant-ph]

(or arXiv:2602.15324v3 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2602.15324 Focus to learn more arXiv-issued DOI via DataCite

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