TY - JOUR
T1 - From Ramanujan graphs to Ramanujan complexes
AU - Lubotzky, Alexander
AU - Parzanchevski, Ori
N1 - Publisher Copyright:
© 2019 The Author(s) Published by the Royal Society.
PY - 2020/1/24
Y1 - 2020/1/24
N2 - 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.
AB - 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.
KW - Complexes
KW - Graphs
KW - Ramanujan
UR - http://www.scopus.com/inward/record.url?scp=85076311831&partnerID=8YFLogxK
U2 - 10.1098/rsta.2018.0445
DO - 10.1098/rsta.2018.0445
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C2 - 31813373
AN - SCOPUS:85076311831
SN - 1364-503X
VL - 378
JO - Philosophical transactions. Series A, Mathematical, physical, and engineering sciences
JF - Philosophical transactions. Series A, Mathematical, physical, and engineering sciences
IS - 2163
M1 - 20180445
ER -