General Antonyms:
* Fail
* Miss
* Neglect
* Abandon
* Forsake
More Specific Antonyms:
* Lose (for a competition or goal)
* Destroy (for a positive accomplishment)
* Undermine (for a positive effect)
* Deter (for a positive outcome)
* Prevent (for a positive result)
The best antonym will depend on the specific meaning of "achieve" in the context of your sentence.