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تسجيل مستخدم جديدSuper-Golden-Gates for PU(2)
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To each of the symmetry groups of the Platonic solids we adjoin a carefully designed involution yielding topological generators of PU(2) which have optimal covering properties as well as efficient navigation. These are a consequence of optimal strong approximation for integral quadratic forms associated with certain special quaternion algebras and their arithmetic groups. The generators give super efficient 1-qubit quantum gates and are natural building blocks for the design of universal quantum gates.
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In a seminal series of papers from the 80s, Lubotzky, Phillips and Sarnak applied the Ramanujan-Petersson Conjecture for $GL_{2}$ (Delignes theorem), to a special family of arithmetic lattices, which act simply-transitively on the Bruhat-Tits trees a
ssociated with $SL_{2}(mathbb{Q}_{p})$. As a result, they obtained explicit Ramanujan Cayley graphs from $PSL_{2}left(mathbb{F}_{p}right)$, as well as optimal topological generators (Golden Gates) for the compact Lie group $PU(2)$. In higher dimension, the naive generalization of the Ramanujan Conjecture fails, due to the phenomenon of endoscopic lifts. In this paper we overcome this problem for $PU_{3}$ by constructing a family of arithmetic lattices which act simply-transitively on the Bruhat-Tits buildings associated with $SL_{3}(mathbb{Q}_{p})$ and $SU_{3}(mathbb{Q}_{p})$, while at the same time do not admit any representation which violates the Ramanujan Conjecture. This gives us Ramanujan complexes from $PSL_{3}(mathbb{F}_{p})$ and $PU_{3}left(mathbb{F}_{p}right)$, as well as golden gates for $PU(3)$.
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