Answer
Zero work but might still need an external force.
Work Step by Step
It requires 0 total work to move from point A to point B. However, it might still require a force to move the particle. Imagine two potential wells with a hill in the middle. To move from one well to the other, some energy must be given to the particle to get it over the hill. However, it can then be removed again as the particle comes down the hill on the other side. So you might need to do positive work to get it going in the right direction and then negative work can be done once you're partway there.
