Hi Maxime,
After some quick reverse engineering of the site with the Chrome Developer Tools, here's the API it is using:
http://www.wikidata.org/w/api.php ?callback=[YOUR CALLBACK NAME] &action=wbsearchentities &format=json &language=en &type=item &continue=0 &_=[TIMESTAMP (as a cache buster)] &search=[YOUR QUERY]
If you don't need the callback, then the API is as follows (note the missing cache buster):
http://www.wikidata.org/w/api.php &action=wbsearchentities &format=json &language=en &type=item &continue=0 &search=[YOUR QUERY]
Hope this helps.
Best, Tom