Anna learnt to play chess at the age of two, taught by her parents, both of whom are professional chess coaches. She played her first tournament at five, and in the same year she placed second in the under-10 girls championship of Lviv Oblast.

Anna won several medals at Ukrainian, European and World Youth Championships. She is a six-time European Youth Champion among girls in different age categories, World Under-16 and World Under-20 Girls Champion. Additionally, many times she was the prize winner of these competitions. At the age of 13, in 2003, Anna won the Ukrainian Women's Championship, repeating this success in 2014. Anna became a Woman FIDE Master in 2001, and a Woman International Master in 2002. Anna was awarded the titles of International Master in 2007 and Grandmaster in 2012.

Anna was the fourth female player, after Judit Polgar, Humpy Koneru and Hou Yifan, to cross the Elo 2600 rating mark, in July 2012