There is no QuickType API but you can get similar functionality with the ACEAutocompleteBar library from GitHub. Just for the record, I have listed the steps to get this working with Swift:
1) Import the files from the folder "ACEAutocompleteBar" to your project.
2) Create a bridging header and write #import "ACEAutocompleteBar.h" at the top.
3) Make your view containing the text field an ACEAutocompleteDataSource and ACEAutocompleteDelegate. 
4) Implement the minimumCharactersToTrigger function
func minimumCharactersToTrigger(inputView: ACEAutocompleteInputView!) -> UInt {
    return 1
}
5) Implement the inputView function.
func inputView(inputView: ACEAutocompleteInputView!, itemsFor query: String!, result resultBlock: (([AnyObject]!) -> Void)!) {
    inputView.hidden = false
    inputView.alpha = 0.75
    if resultBlock != nil{
      dispatch_async(dispatch_get_global_queue(DISPATCH_QUEUE_PRIORITY_DEFAULT, 0)) {
        var data:NSMutableArray = []
        if(self.myTextField.isFirstResponder()){    
          //query your data source with 'query' and add any of the results to data to pass back                
        }
        dispatch_async(dispatch_get_main_queue()) {resultBlock(data as [AnyObject])}  
      }
   }
}
6) Implement the textField function.
func textField(textField: UITextField!, didSelectObject object: AnyObject!, inInputView inputView: ACEAutocompleteInputView!) {
    textField.text = String(object)
}
7) Call setAutocompleteWithDataSource on the textField.
self.myTextField.setAutocompleteWithDataSource(self, delegate: self, customize: {
        inputView in
        // customize the view (optional)
        inputView.font = UIFont.systemFontOfSize(20)
        inputView.textColor = UIColor.whiteColor()
        inputView.backgroundColor = UIColor.blueColor()
        inputView.hidden = false
    })
Hope that helps anyone looking for this kind of functionality!