Most libertarians oppose capital punishment. They argue that capital punishment is an extreme exertion of state power, it is contrary to the values of a free society, authoritarian countries tend to be retentionist, and liberal-democratic societies generally abolitionist.
Proponents of capital punishment believe that such punishment may be justified as a deterrent to particularly atrocious crimes and as a means of keeping dangerous individuals permanently incapacitated. Furthermore, if people commit crimes, they may sacrifice their rights and if the legal system is legitimate, perhaps capital punishment is justified. The U.S. Libertarian Party, a right-libertarian American third party, opposes "the administration of the death penalty by the state" despite the large stake that conservatives would have in abolishing the death penalty.
There is also the fundamental problem of the possibility of error or even the outright framing of the accused. To that effect, lack of trust in government to make decisions (including life-and-death decisions) competently or for the best motives may confound the issue; already deeply distrustful of government, they say, it should not be trusted to be an arbiter of life and death without error.