Strictly speaking, only defined for decision problems: cs.stackexchange.com/questions/9664/is-it-necessary-for-np-problems-to-be-decision-problems/128702#128702
Interesting because of the Cook-Levin theorem: if only a single NP-complete problem were in P, then all NP-complete problems would also be P!
We all know the answer for this: either false or independent.
A problem such that all NP problems can be reduced in polynomial time to it.
This is the most interesting class of problems for BQP as we haven't proven that they are neither:
- P: would be boring on quantum computer
- NP-complete: would likely be impossible on a quantum computer
P for quantum computing!
Heck, we know nothing about this class yet related to non quantum classes!
- conjectured not to intersect with NP-complete, because if it were, all NP-complete problems could be solved efficiently on quantum computers, and none has been found so far as of 2020.
- conjectured to be larger than P, but we don't have a single algorithm provenly there:
- it is believed that the NP complete ones can't be solved
- if they were neither NP-complete nor P, it would imply P != NP
- we just don't know if it is even contained inside NP!
- math.stackexchange.com/questions/361422/why-isnt-np-conp "Why isn't NP = coNP?"
- stackoverflow.com/questions/17046440/whats-the-difference-between-np-and-co-np
- cs.stackexchange.com/questions/9795/is-the-open-question-np-co-np-the-same-as-p-np
- mathoverflow.net/questions/31821/problems-known-to-be-in-both-np-and-conp-but-not-known-to-be-in-p
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