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Anderson, one of three unseeded players remaining in the men's field of the combined ATP Masters and WTA tournament, reached the quarter-finals of an elite Masters event for the second time.
Anderson, who had surgery in January to remove bone chips from his right elbow, had already ousted world No 4 David Ferrer in the second round.
VIDEO: Anderson v Simon, highlights
In the quarter-finals, he'll face either sixth-seeded Czech Tomas Berdych or 10th-seeded Richard Gasquet of France.
Wednesday's fourth-round matches sees the "big four" of men's tennis in action.
World No 1 and top seed Novak Djokovic faces American Sam Querrey - the last man to beat the Serbian star back on October 31.
Since then Djokovic has put together a 20-match winning streak and won a fourth career Australian Open title as well as the ATP title in Dubai.
Djokovic didn't attach much importance to the fact that Querrey beat him in his opening match at the Paris Masters last year, noting that the indoor surface was very different from the conditions they would encounter on the hard courts of the California desert.
"I won four times against him," said Djokovic, who indeed leads their head-to-head 4-1. "Yes, he has won the last encounter indoors - different circumstances and conditions."
"I mean, we'll play in front of his crowd and he has a big serve and big game and he can come up with the goods when needed," Djokovic added, noting that against such a dangerous server he couldn't afford to start as slowly as he did in a 7-6 (7/4), 6-1 third-round victory over Belgian Grigor Dimitrov.
Defending champion and second seed Roger Federer will battle fellow Swiss Stanislas Wawrinka for a quarter-final spot.
Federer, a four-time winner of the Indian Wells title, is bidding for his first tournament win of 2013 having been denied in Rotterdam and Dubai.
World No 3 Andy Murray faces unseeded Argentinian Carlos Berlocq while Rafael Nadal, ranked fifth in the world as he continues his return from a seven-month injury absence faces a potentially tricky fourth-round match against qualifier Ernests Gulbis.
The 24-year-old Latvian is on a hot streak. He became the first qualifier to win an ATP title this year when he triumphed at Delray Beach prior to Indian Wells.
Nadal, whose comeback tour of three Latin American clay court tournaments saw him reach one final and win two titles, has played just one match this week. After a first round bye he beat American Ryan Harrison, then advanced on a walkover when Leonardo Mayer withdrew from their third-round match.
Other fourth-round matches include eighth-seeded Jo-Wilfried Tsonga of France against Canadian Milos Raonic in a rematch of their record-breaking first-round clash at the London Olympics, which Tsonga won 6-3, 3-6, 25-23.
Two women's quarter-finals were due on court, including a rematch of last year's French Open final between Maria Sharapova and Sara Errani.
Russia's Sharapova defeated the Italian at Roland Garros to complete a career Grand Slam.
Former Wimbledon champion Petra Kvitova, the fifth seed, faces Russian Maria Kirilenko for a semi-final spot.
