That's such a silly question. As if it could matter what I think about whether something *should* exist when it already does exist. What if someone said "no" it shouldn't -- what could ever result from that other than a feigned attempt to reduce the number of such devices. It is as of now, totally impossible to rid the world of every single nuclear device in existence. Good luck with that.
The hydrogen bomb should exist because if multiple major powers in the world have them, there is mutually shared destruction. The means that no country can use their hydrogen bomb because if they deployed it, other countries would as well. This would result in the whole world being blown up, so countries will never do this.
The hydrogen bomb is an extremely perilous weapon of mass destruction. No person should have the power to kill so many people in one fell swoop. When in the wrong hands, the results of hydrogen bombs are disastrous. However, even in the right hands, hydrogen bombs can do no good. They only serve to kill, which is universally unjustified and immoral.
I don't think the hydrogen bomb should exist because we should not be developing mass weapons. History shows that destruction and loss is the only thing that comes from mass weapons such as the hydrogen bomb. The bombing of Hiroshima with the atomic bomb shows caused destruction because world leaders relied on violence instead of negotiating peace. Instead of spending resources on weapons like hydrogen bombs, those resources should go to developing peaceful relations between countries to prevent the need for something like a hydrogen bomb.