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from the just-when-you-thought-you-knew-how-things-worked dept.
Major quantum computational breakthrough is shaking up physics and maths:
MIP* = RE is not a typo. It is a groundbreaking discovery and the catchy title of a recent paper in the field of quantum complexity theory. Complexity theory is a zoo of "complexity classes" – collections of computational problems – of which MIP* and RE are but two.
The 165-page paper shows that these two classes are the same. That may seem like an insignificant detail in an abstract theory without any real-world application. But physicists and mathematicians are flocking to visit the zoo, even though they probably don't understand it all. Because it turns out the discovery has astonishing consequences for their own disciplines.
[...] RE stands for problems that can be solved by a computer. It is the zoo. Let's have a look at some subclasses.
[...] Allowing the provers of MIP to share an entangled qubit leads to the class MIP*.
It seems obvious that communication between the provers can only serve to help the provers coordinate lies rather than assist the interrogator in discovering truth. For that reason, nobody expected that allowing more communication would make computational problems more reliable and solvable. Surprisingly, we now know that MIP* = RE. This means that quantum communication behaves wildly differently to normal communication.
IP = Interactive Polynomial time solvable using an interactive proof.
MIP = Multiple Interactive Provers that are not allowed to communicate between each other.
RE = Recursively Enumerable.
[Edited to add a definition for RE and to correct the definition of IP. - Ed]
(Score: 2, Interesting) by Anonymous Coward on Sunday August 23 2020, @08:03AM
RE stands for recursively enumerable, it's the set of all computable functions and encompasses every other complexity class. MIP* is new to me so I may be wrong about this but here's my understanding of it.
One of the definitions of NP is that a decision problem in NP is verifiable in polynomial time, in your maze example it may take a long time to find a path through through the maze but if someone gives you a path it's easy to check whether it's correct (just follow it and see if it leads you out).
Some problems could be a lot harder to check than that though, in a sense these are harder still than NP as even if you're given the answer there's no easy way to verify it's correct. A solution to this is to come up with some heuristics that you can have the algorithm output along with the answer and these help reassure you the answer is valid, from my understanding thats IP.
MIP is an extension of this where you use multiple methods of finding a solution and cross check the results against each other which gives you more power than checking everything yourself. Of course you want these 2 systems to be independent otherwise an error in one could spread to the other and mess up your cross checking.
This seems to be saying that in the quantum world you actually get more power by allowing the different methods to share information, in fact you get so much more power that the set of problems that you can verify is equal to all functions that can be computed (MIP* = RE).
The reason this is so big is that it proves that if something can be computed *at all* then it's also possible to verify the solution is correct with less work than it took to prove it.