I think you will find that the number of people found NOT GUILTY of a previous conviction for murder on a retrial is greater than the number of people who are actually executed following a conviction for murder. These are American stats, feel free to check them.
This means that the legal process is so screwed up that more people are unjustly found guilty of murder than are actually executed for murder. Fortunately miscarraiges of justice are usually stopped before the execution.
However, the stats mean you would have to be a fool to assert that no innocent person is ever murdered by the legal system.
As there are viable alternatives to execution that give some possibility of mistakes being rectified (i.e. prison), there is really no execuse for judicial killing.
You may feel it provides justice, but it also causes injustice. Think on it, sat in old sparky, wating for the end, knowing you're not guilty. Say to me you'd be willing to die like that and I can take you approving of the death penalty seriously - if I believe you.
Also, all a mandatory death sentence for child molestors would do is make sure more of them kill children, as they have nothing to lose. Thus a commendable desire for justice can actually turn out to cause even more suffering for those it is intended to protect.