In a small sauce pan, combine sugar and water. Heat on medium heat until the sugar has completely dissolved.
Take off the heat and pour simple syrup into a glass jar.
Add ginger and rosemary to the jar and cover with the lid. This will help the ginger and rosemary flavors infuse.
Infuse for at least 1 hour. Strain the syrup to get the ginger and rosemary pieces out.
Store your simple syrup in the fridge. It will keep for 1-2 weeks.
Combine vodka, soda, lime juice, and syrup in a copper mug. Stir and garnish with lime and rosemary.