![]() A 4×4 braiding matrix Bt defines the braided tensor product. The Hopf algebra is U(gl(1|1)), the Universal Enveloping Algebra of the gl(1|1) superalgebra. The framework is that of a graded Hopf algebra endowed with a braided tensor product. The multiparticle sectors of N, braided, indistinguishable Majorana fermions are constructed via first quantization. While we outline the theoretical basis for such a platform, details on the physical implementation remain open.Ī Z2-graded qubit represents an even (bosonic) “vacuum state” and an odd, excited, Majorana fermion state. A concrete encoding on these tiling spaces of topological quantum information processing is also presented by making use of inflation and deflation of such tiling spaces. In particular, we study the correspondence between the fusion Hilbert spaces of the simplest non-abelian anyon, the Fibonacci anyons, and the tiling spaces of the one-dimensional Fibonacci chain and the two-dimensional Penrose tiling quasicrystals. Different from anyons, quasicrystals are already implemented in laboratories. We propose quasicrystal materials as such a natural platform and show that they exhibit anyonic behavior that can be used for topological quantum computing. A topological quantum computing platform promises to deliver more robust qubits with additional hardware-level protection against errors that could lead to the desired large-scale quantum computation. ![]() ![]() The concrete realization of topological quantum computing using low-dimensional quasiparticles, known as anyons, remains one of the important challenges of quantum computing.
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