We propose an encoding for topological quantum computation utilizing quantum representations of mapping class groups. Leakage into a non-computational subspace seems to be unavoidable for universality. We are interested in the possible gate sets which can emerge in this setting. As a first step, we prove that for abelian anyons, all gates from these mapping class group representations are normalizer gates. Results of Van den Nest then allow us to conclude that for abelian anyons this quantum computing scheme can be simulated efficiently on a classical computer. With an eye toward more general anyon models we additionally show that for Fibonnaci anyons, quantum representations of mapping class groups give rise to gates which are not generalized Clifford gates.
We study the stated skein modules of marked 3−manifolds. We generalize the splitting homomorphism for stated skein algebras of surfaces to a splitting homomorphism for stated skein modules of 3−manifolds. We show that there exists a Chebyshev-Frobenius homomorphism for the stated skein modules of 3-manifolds which extends the Chebyshev homomorphism of the skein algebras of unmarked surfaces originally constructed by Bonahon and Wong. Additionally, we show that the Chebyshev-Frobenius map commutes with the splitting homomorphism. This is then used to show that in the case of the stated skein algebra of a surface, the Chebyshev-Frobenius map is the unique extension of the dual Frobenius map (in the sense of Lusztig) of O q 2 (SL(2)) through the triangular decomposition afforded by an ideal triangulation of the surface. In particular, this gives a skein theoretic construction of the Hopf dual of Lusztig's Frobenius homomorphism. A second conceptual framework is given, which shows that the Chebyshev-Frobenius homomorphism for the stated skein algebra of a surface is the unique restriction of the Frobenius homomorphism of quantum tori through the quantum trace map.
We make two related observations in this paper. First the representations of mapping class groups from the Ising TQFT and its quantum group counterpart SU (2) 2 are neither equivalent as representations nor Galois conjugate to each other. Hence mapping class group representations obtained from quantum skein theory are fundamentally distinct from those obtained from quantum group Reshetikhin-Turaev or geometric quantization constructions. Then we generalize the asymptotic faithfulness of the skein quantum SU (2) representations of mapping class groups of orientable closed surfaces to skein quantum SU (3). We conjecture asymptotic faithfulness holds for skein quantum G representations when G is a simply-connected simple Lie group. The difficulty for such a generalization lies in the lack of an explicit description of the fusion spaces with multiplicities to define an appropriate complexity of state vectors.
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