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!
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