Because we are not God, we do not have the right nor the ability to judge, condemn, and punish others for their sins.
The state, however, has the right and the obligation to protect us all from each other. It is a right and an obligation that we have given it, collectively. Thus, although the state does not have a right to judge, condemn and punish people for their sins, it does have the right to judge, condemn and respond to our criminal actions in the interest of public safety and security.
I believe that in the interest of public security the state does have the right to end the lives of human beings who show themselves to be an ongoing threat to the lives of others. Serial killers, serial kidnappers, torturers, and rapists, terrorists, and people who kill while incarcerated are, in my mind, people who have shown us that they will be an ongoing threat to the lives of others. And as such, I do not believe that society has any obligation to accept that threat in keeping them alive.
This particular killing is borderline. It's possible that the killer suffers from some treatable mental condition. And once treated, would pose no further obvious threat. However, if such could not be shown to be the case, the random opportunistic nature of the crime, and the irrationality of it's reasoning would indicate, to my mind, that this man does pose an ongoing threat that society is not obliged to accept. And thus would warrant his execution.