In Toaru Majutsu no Index, the city can be brought into existence through certain spells based off the Cthulhu Mythos.
According to the story, when certain conditions are met, the city would rise to the surface from the bottom of the ocean. When R'lyeh surfaced, it was said that the weak-minded would go insane while poets and artists would receive strange inspiration. These conditions include the positions of the stars and submarine volcanic activity. With the rising of the city would also come the emergence of the entity Cthulhu.
As with other spells based on the Cthulhu Mythos, spells involving the city are classed as creating something from nothing - calling into existence something which shouldn't exist. The pages of the Necronomicon used for a ceremony to draw up the city are 45, 90-120, 200 and 210-222.
On August 2nd, three years before the current year of the story timeline, the magic cabal Dusk Waiting to Awaken carried out an experiment in the Pacific involving summoning R'lyeh in order to acquire inspiration and information to overcome a dead end in their research. The ceremony caused 103 people aboard the resource extraction ship Condor to go insane and would have wiped out the residents of every island within 1000km, had the experiment not failed. After the incident and the discovery of magic symbols on the ship, England installed a facility in the area to keep an eye on the location where the ceremony took place.
Necessarius Special Admission Test SS
After obtaining the Necronomicon, Arlands Darkstreet of Dusk Waiting to Awaken took it to an ocean salvage terminal in the Strait of Dover, in order to perform a ceremony to summon R'lyeh and obtain the inspiration his cabal needed. Though the terminal was far from the location where R'lyeh was sunk, Arlands and his men were able to recreate the conditions of the sunken city on it, using lighting and observational equipment to simulate the non-Euclidean geometry of the city and exploding fire ships to simulate the earthquakes heralding the surfacing.
As a result, when the ceremony was carried out, the terminal became the equivalent of R'lyeh itself, with Cthulhu partially appearing before the Amakusa managed to make the evil god leave by interfering with the artificially created conditions which created the twisted landscape synonymous with the sunken city.