Xperia Blog has consolidated several reports of Sony Xperia Z Ultra owners complaining about issues with the display. Most of the problems seem to happen when trying to unlock the phone, with the display just refusing to switch on. A video notes that the backlight switches on as well, and one can hear the underlying OS sounds as you tap on the screen - it's just the display that stays off.
A user at XDA forum notes similar experience when he tried to press the power button of his Xperia Z Ultra. The backlight turned on but the screen remained completely blank. He states that the Android runs fine underneath and when pressing back, home, app or even volume control buttons the device gives audio feedback. However the display remains blank and the device is unusable. He also notes that the screen turned on only after pressing the power key anywhere between 5 and 150 times, indicating the random nature of the problem.
Xperiablog notes that several other users on XDA developers forum and YouTube have voiced their complaints regarding the phone's display. The report also notes that some display show green lines when an attempt is made to unlock the device. There is no official comment from Sony on the issue till now.