There is a reason we have rules concerning 'due process', 'innocent until proven guilty', and 'cruel and unusual punishment'.
Summary execution violates each of them.
State complicity in mob-'justice' (ie. releasing him into the general prison population and making it known that the guards will not interfere, as suggested earlier in this thread) violates at least two of them, and probably all three.
Both of those are evil.
If a man walked into a theater you and your child were seeing a film in and murdered your child, then was caught, would you ask the police to give him a trial and full investigation? Some people would, in this day and age. I wouldn't. If it was my child, I would want to return the favor myself. I will agree that there is a fine line between necessary punishment and mob-fed verdict.
If anything, the killer is lucky to still be alive. The police could have easily just shot him and this would be over. Of course, known child-killers are generally handled in prison, as mentioned previously.