Q: What Causes Error Code 4013?

From: Get help with iOS update and restore errors - Apple Support.

Try these steps to restore your iOS device:

1. Install the latest version of iTunes.
2. Restart your computer.
3. Check for updates to your Mac or PC. If you restart your computer for an update, check for updates again after you restart.
4. Try to restore your iOS device using another USB cable.
5. Try to restore your iOS device on another computer.

If you continue to see error 4005, 4013, or 4014 when you try to restore your iPhone, iPad, or iPod touch, contact us.

Then click on that last link - Contact Us.

As Kappy said

- If still not successful that usually indicates a hardware problem and an appointment at the Genius Bar of an Apple store is in order.

Apple Retail Store - Genius Bar