Universal quantum computing and three-manifolds
| Autor: | Planat, Michel, Aschheim, Raymond, Amaral, Marcelo M., Irwin, Klee |
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| Rok vydání: | 2018 |
| Předmět: | |
| Zdroj: | Symmetry (MDPI) 10 (12), 773 (2018) |
| Druh dokumentu: | Working Paper |
| Popis: | A single qubit may be represented on the Bloch sphere or similarly on the $3$-sphere $S^3$. Our goal is to dress this correspondence by converting the language of universal quantum computing (UQC) to that of $3$-manifolds. A magic state and the Pauli group acting on it define a model of UQC as a positive operator-valued measure (POVM) that one recognizes to be a $3$-manifold $M^3$. More precisely, the $d$-dimensional POVMs defined from subgroups of finite index of the modular group $PSL(2,\mathbb{Z})$ correspond to $d$-fold $M^3$- coverings over the trefoil knot. In this paper, one also investigates quantum information on a few "universal" knots and links such as the figure-of-eight knot, the Whitehead link and Borromean rings, making use of the catalog of platonic manifolds available on the software SnapPy. Further connections between POVMs based UQC and $M^3$'s obtained from Dehn fillings are explored. Comment: 17 pages, 5 figures, 6 tables introduction much improved |
| Databáze: | arXiv |
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