At the moment, there's no conclusive evidence linking a high sugar diet with acne. Some studies with small sample sizes (~50 participants each) have found a correlation between people who eat a lot of sugar and the appearance of pimples, but no studies have been done to test this correlation - to do so, you would have to give a group of people a lot of sugar for a long period of time and another group very little sugar, and see what happens to their skin. This would probably be unethical because a high sugar diet is unhealthy for several other reasons.

Acne is caused by hormones triggering the overproduction of cells in the pores in our skin, which causes those pores to become blocked, and traps bacteria inside which are able to grow. The growth triggers pus formation, which is the white substance that often fills pimples. Teenagers are more susceptible to acne because they begin producing hormones in large quantities, which exacerbates this process. It's possible that diet can alter the amount of hormones circulating through the body, but we don't know yet if this is enough to cause acne.

Reducing sugar in the diet is often recommended to treat acne because there's no downside to trying! A low sugar diet is much healthier for several reasons, so its worth a shot.