In a food processor, process the rosemary and sugar until fully combined, about 1 minute—there will be visible pieces of rosemary. DO AHEAD: Rosemary crystals can be prepared in advance and stored, in an airtight container at room temperature, up to 2 weeks.
In a small saucepan, bring the rose syrup and distilled or tap water to a boil over high heat. Reduce the heat to moderate and simmer, uncovered, until reduced by half, 12 to 15 minutes. DO AHEAD: Rose elixir can be prepared in advance and refrigerated up to 2 weeks.
Pour the lime juice onto a small plate and spread the rosemary crystals on a second small plate. Dip the rim of a 12-ounce glass into the lime juice, then dip it into the rosemary crystals to lightly coat. Repeat with a second 12-ounce glass.
Pour 1 ounce each of rose elixir, yuzu juice, and simple syrup into each prepared glass.
