Structure of 2D Topological Stabilizer Codes

We provide a detailed study of the general structure of two-dimensional topological stabilizer quantum error correcting codes, including subsystem codes. Under the sole assumption of translational invariance, we show that all such codes can be understood in terms of the homology of string operators that carry a certain topological charge. In the case of subspace codes, we prove that two codes are equivalent under a suitable set of local transformations if and only they have equivalent topological charges. Our approach emphasizes local properties of the codes over global ones.Comment: 54 pages, 11 figures, version accepted in journal, improved presentation and result

The amyloid precursor protein controls PIKfyve function

While the Amyloid Precursor Protein (APP) plays a central role in Alzheimer's disease, its cellular function still remains largely unclear. It was our goal to establish APP function which will provide insights into APP's implication in Alzheimer's disease. Using our recently developed proteo-liposome assay we established the interactome of APP's intracellular domain (known as AICD), thereby identifying novel APP interactors that provide mechanistic insights into APP function. By combining biochemical, cell biological and genetic approaches we validated the functional significance of one of these novel interactors. Here we show that APP binds the PIKfyve complex, an essential kinase for the synthesis of the endosomal phosphoinositide phosphatidylinositol-3,5-bisphosphate. This signalling lipid plays a crucial role in endosomal homeostasis and receptor sorting. Loss of PIKfyve function by mutation causes profound neurodegeneration in mammals. Using C. elegans genetics we demonstrate that APP functionally cooperates with PIKfyve in vivo. This regulation is required for maintaining endosomal and neuronal function. Our findings establish an unexpected role for APP in the regulation of endosomal phosphoinositide metabolism with dramatic consequences for endosomal biology and important implications for our understanding of Alzheimer's disease