Jump in water and pull out an item (either a sword or a Deku stick) just as Link hits the water. The next time Link uses that item it will appear in his hand without playing the unsheathing animation.
It is used in this video to cucco jump with a Deku Stick, which is not normally possible.
A much more prominent use for Quick Draw is Duplicating a Bottle over your Sword
By pressing B on while shielding a damage source that has a multiple-frame hitbox, or by shielding damage from two damage sources in succession, you can cancel the drawing of the sword. This causes Quick Draw to occur when you next pull the sword. More interestingly, this also sets the "last button used" value to your B button, while still keeping the previous item you were holding in your hands. This allows you to Shield Swipe over the sword for Bottle B. A bomb or bombchu is generally used due to its availability. As an adult, there is one frame to press B and you must be in front of wall to get Quick draw. As a child, there are multiple frames to press B and you do not need to be near a wall, so it is considerably easier.
As an adult:
As a child: