An example of universal braiding in Links--Gould topological quantum computation

Abstract

We construct a universal quantum computational encoding via braiding in Links--Gould fusion spaces associated to $U_q^H(\mathfrak{sl}(2|1))$ where $q$ is on the unit circle and is not a root of unity. The construction uses only neglectons and is realized in the morphism spaces $\mathrm{Hom}_{U_q^H}(V(0,α),V(0,-\frac{1}{2})^{\otimes 2N}\otimes V(0,α)).$ Under explicit positivity and arithmetic constraints these spaces admit a $2^N$-dimensional computational subspace with a positive definite Hermitian form. We prove single-qubit density in $\mathrm{U}(2,1)$ and approximate arbitrary unitary operators on $N$-qubit computational subspaces with arbitrarily small leakage. The calculations use explicit inclusions, projections, and Hermitian pairings without requiring $F$-matrices.