In Brad Ziegler's brief major-league pitching career, he has not given up a run. Period. According to this news article:

Athletics rookie reliever Brad Ziegler stretched his scoreless-inning streak to 30 - extending the major league record to begin a career...

The prior streak of scoreless innings pitched to begin a career is 25, dating back to 1907. Heck, the Cubs have even won a World Series championship more recently than that!

The overall record for scoreless innings pitched is, of course, 59 by the Dodgers' Orel Hershiser in 1988.