The 28-year-old is Bristol's first signing for next term, building towards new head coach Pat Lam's arrival from Connacht in the summer.
Former Hurricanes centre Leiua recovered from two knee ligament operations last term, and has had to contend with an increasingly crowded midfield picture this year at Wasps.
"We're delighted to beat off interest from a number of clubs and capture Alapati's signing," said Bristol head coach Mark Tainton.
