Quantum Provers to Infinity and Beyond

The Internets are buzzing about the new paper MIP* = RE by Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright and Henry Yuen. See posts by Scott, Boaz, not to mention a wonderful backstory by Vidick himself and a tweet stream by Yeun. I'm not an expert enough to verify or even try to explain the proof so I'll just give a brief overview of the result.

For those not familiar with the classes, RE (recursively enumerable) is the simplest of all complexity classes, a language is in RE if there is some Turing machine M such that x is in L if and only if M on input x accepts. For x not in L, M on x can reject or run forever. The classic halting problem, the set of descriptions of Turing machines that halt on empty input, is RE-complete. To nitpick the notation, it should have been r.e. and even c.e. (computably enumerable), a more standard notation these days. But given the importance of the result, we can give the authors a pass.

MIP* is the set of things provable to a classically random polynomial-time verifier by two separated provers with an unlimited number of quantumly entangled qubits. Without the quantum entanglement, MIP = NEXP, nondeterministic exponential time, and last year Natarajan and Wright showed that MIP* could do at least exponentially better in their paper, NEEXP in MIP*. NEEXP seems large but still only consists of computable sets. RE gets outside of the computable realm.

I found the first paper more surprising, as it showed that quantum entanglement actually gets more, much more, than classical provers. The second paper does get a much stronger and tight result, and still highly surprising in its own right, as it requires disproving the Connes' embedding conjecture. In the end we may just consider this one result, as the second paper subsumes the first both in theorem and authors.

We didn't award the 2019 theorem of the year to Natarajan and Wright, instead opting for a paper that had more, how should I say this, sensitivity. This new paper is certainly the front runner for the 2020 honors, albeit it is only mid-January.