# Consequences of SAT ∈ BQP

"Quantum magic won't be enough" (Bennett et al. 1997)

If you throw away the problem structure, and just consider the space of $2^n$ possible solutions, then even a quantum computer needs about $\sqrt{2^n}$ steps to find the correct one (using Grover's algorithm) If a quantum polynomial time algorithm for an $\text{NP}$-complete problem is ever found, it must exploit the problem structure in some way.

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I've some (basic) questions that no one seems to have asked so far on this site (maybe because they are basic). Suppose someone finds a bounded error quantum polynomial time algorithm for $\text{SAT}$ (or any other $\text{NP}$-complete problem), thus placing $\text{SAT}$ in $\text{BQP}$, and implying $\text{NP} \subseteq \text{BQP}$.

## Questions

Which would be the theoretical consequences of such a discovery? How would the overall picture of complexity classes be affected? Which classes would become equal to which others?

Source

• It seems that at least three different questions may be prepared starting from this one: (a) Suppose someone [...] -> consequences for CS (b) Suppose someone [...] -> consequences for Physics (c) the two final questions on classical vs quantum "structure-exploitation". I suggest this splitting in 3 would (eventually) produce higher-quality questions and meaninful answers. Apr 10 '18 at 14:49
• For all three, I also recomment checking the phrase about "bullet 2 would be contradicted", which looks like a residual from a context with bullets. Apr 10 '18 at 16:05
• That's weird. The book you linked seems to copy Theoretical Computer Science Q and A's verbatim: see cstheory.stackexchange.com/questions/6154/… . Is it even legal to have such a thing behind a paywall? Apr 10 '18 at 17:50
• I guess it is legal if attribution is proper, but that is hard to check if stuff is behind a paywall. Apr 10 '18 at 17:56