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Register a new userRamanujan complexes and Golden Gates in PU(3)
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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 associated 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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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.
Ramanujan graphs are graphs whose spectrum is bounded optimally. Such graphs have found numerous applications in combinatorics and computer science. In recent years, a high dimensional theory has emerged. In this paper these developments are surveyed. After explaining their connection to the Ramanujan conjecture we will present some old and new results with an emphasis on random walks on these discrete objects and on the Euclidean spheres. The latter lead to golden gates which are of importance in quantum computation.
The total-variation cutoff phenomenon has been conjectured to hold for simple random walk on all transitive expanders. However, very little is actually known regarding this conjecture, and cutoff on sparse graphs in general. In this paper we establish total-variation cutoff for simple random walk on Ramanujan complexes of type $tilde{A}_{d}$ ($dgeq1$). As a result, we obtain explicit generators for the finite classical groups $PGL_{n}(mathbb{F}_{q})$ for which the associated Cayley graphs exhibit total-variation cutoff.
This paper is concerned with a class of partition functions $a(n)$ introduced by Radu and defined in terms of eta-quotients. By utilizing the transformation laws of Newman, Schoeneberg and Robins, and Radus algorithms, we present an algorithm to find Ramanujan-type identities for $a(mn+t)$. While this algorithm is not guaranteed to succeed, it applies to many cases. For example, we deduce a witness identity for $p(11n+6)$ with integer coefficients. Our algorithm also leads to Ramanujan-type identities for the overpartition functions $overline{p}(5n+2)$ and $overline{p}(5n+3)$ and Andrews--Paules broken $2$-diamond partition functions $triangle_{2}(25n+14)$ and $triangle_{2}(25n+24)$. It can also be extended to derive Ramanujan-type identities on a more general class of partition functions. For example, it yields the Ramanujan-type identities on Andrews singular overpartition functions $overline{Q}_{3,1}(9n+3)$ and $ overline{Q}_{3,1}(9n+6)$ due to Shen, the $2$-dissection formulas of Ramanujan and the $8$-dissection formulas due to Hirschhorn.
Using the $q$-Wilf--Zeilberger method and a $q$-analogue of a divergent Ramanujan-type supercongruence, we give several $q$-supercongruences modulo the fourth power of a cyclotomic polynomial. One of them is a $q$-analogue of a supercongruence recently proved by Wang: for any prime $p>3$, $$ sum_{k=0}^{p-1} (3k-1)frac{(frac{1}{2})_k (-frac{1}{2})_k^2 }{k!^3}4^kequiv p-2p^3 pmod{p^4}, $$ where $(a)_k=a(a+1)cdots (a+k-1)$ is the Pochhammer symbol.
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