Playing her first tournament since January’s Australian Open because of a foot injury, Boulter received a first-round bye as the 25th seed.
Former Wimbledon champion Rybakina won the first seven games in taking the first set in 26 minutes, but Boulter regrouped in the second and broke Kazakhstan’s Rybakina to claim a 4-3 lead before serving for the set at 5-4.
Although Boulter saved two break points, Rybakina took advantage of a third to level it up and, after holding serve, broke Boulter thanks to a double fault on match point.
Rybakina will face Russian ninth seed Mirra Andreeva, who beat Denmark’s Clara Tauson 6-3 6-0, in the last 16.
Defending champion Iga Swiatek – chasing a third title in this tournament – dominated Ukraine’s Dayana Yastremska 6-0 6-2.
Poland’s Swiatek will play Karolina Muchova, a 7-5 6-1 winner against fellow Czech Katerina Siniakova, next.
American fourth seed Jessica Pegula beat China’s Wang Xinyu 6-2 6-1 to set up a meeting with Ukrainian Elina Svitolina, who beat American Danielle Collins 6-2 6-4.
