
MIP*=Re - soohyung
https://arxiv.org/abs/2001.04383
======
yifanlu
From Scott Aaronson

> Yet another reason to be excited about this result—one that somehow hadn’t
> occurred to me—is that, as far as I know, it’s the first-ever fully
> convincing example of a non-relativizing computability result.

This is one of the three known barriers any proof of P vs NP must overcome.
Not saying any of this applies to P vs NP but the proof technique demonstrated
is important for such a proof.

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sddfd
Could someone explain what this means? I read it as proof that quantum
computers cannot compute more functions than Turing machines, is that correct?

~~~
yifanlu
It’s a highly technical problem that doesn’t have much practical use.
Essentially it says that a weak (polynomial time) classical computer (called
the “verifier”) with the ability to query (polynomial number of times) a small
number of all-powerful computers (called the “provers”) with unknown/unbounded
computing power is able to be convinced that the prover is indeed all powerful
(convinces the verifier that it knows the answer to some problem that is known
to be difficult). These specific provers can not communicate with each other
(collude) but can share some entangled bits (the quantum part). The result
here is (basically) that the verifier can be convinced that the prover knows
the answer to the halting problem.

The result is more interesting with some context though. One of the biggest
discoveries of the 21st century is that PSPACE=IP. That’s where a single all
powerful prover can convince a weak verifier of any solution in PSPACE (an
extension of P that we believe to be more powerful than P). Then more recently
we found MIP=NEXP which is a bunch of classic all powerful provers (with no
collusion) can convince a single weak verifier of any problem in NEXP (which
is like NP but with much larger proof sizes, thought to be more powerful than
NP). The surprising fact is that multiple all-powerful provers can convince a
verifier of harder problems. Intuitively this doesn’t make sense because we
know each provers are all powerful and are permitted to do anything, even
physically impossible tasks, so the fact that a single prover cannot convince
a verifier of any problems harder than PSPACE is surprising. More surprising
is that multiple all powerful provers can somehow prove harder problems when
it feels like we’re not adding any extra power (they’re already “all
powerful”).

So in that context we find the also surprising fact that if we weaken the “no
collusion” requirement to “only quantum entanglement” and suddenly they can
solve the halting problem (but still no more!).

This line of proving both upper and lower bounds about complexity classes is
exactly what we want to see in the P vs NP world (which is in many ways less
“powerful” of classes but more “practical”). But we don’t even know if P ?=
NEXP.

~~~
joker3
NEXP is nondeterministic exponential time, right? In that case we know that P
/= EXPTIME by the time hierarchy theorem and EXPTIME is contained in NEXP. We
also have a direct result that NP /= NEXP (reference given in
[https://complexityzoo.uwaterloo.ca/Complexity_Zoo:N#nexp](https://complexityzoo.uwaterloo.ca/Complexity_Zoo:N#nexp)).

