There will be several different ways to do it with, as was mentioned, the caveat that occasionally smartthings goes bonkers and might open or close the gate unexpectedly. You will see a number of reports in the forums from people talking about a light coming on expectedly or even a garage door opening. 
If you’re OK with that and you want to just to start exploring, take a look in the community – created wiki on the quick browse lists in the project report section and look for the list for gates. Some of those will just be looking for locks, but there are a couple that wanted to do automatic openers.
There’s one more option which is really simple to use and won’t require any wiring, but it’s expensive. There’s a little tiny actuator intended to work as a robot finger specifically for pushing a button and it has an IFTTT channel so you can get integration with SmartThings that way. But it costs about $89 for the controller and another $49 for the button pusher, so it may be more than you want to spend. I do use them at my house, but I’m quadriparetic so button pushing is a big issue for me.
You can find recent discussion of the microbot option in the following thread about an old air conditioner: