Local officials said the explosions happened in a local market in the city of Bamiyan, the center of Bamiyan province, Tolo News reported.
No group has so far claimed responsibility for the blasts.
Bamiyan is one of the most secure provinces in the city that is visited by thousands of tourists every year. This is the first time that such explosions are happening in the province.
Violence remains high in the country despite efforts for peace.