For the ginger simple syrup:
1) Combine the ginger, water and sugar in a small saucepan and simmer until the sugar has completely dissolved. Set aside to cool.
To make up the cucumber lychee sake:
1) Peel and half the cucumbers. Cut half a cucumber into thin slices. Place the remaining cucumber into a blender and puree.
2) In a cocktail shaker, combine the sake, the pureed cucumber, the ginger simple syrup and lychee juice. Shake to combine the ingredients and serve in ice filled glasses garnished with lychees.