QIP = PSPACE

A great new result in quantum complexity, Rahul Jain, Zhengfeng Ji, Sarvagya Upadhyay and John Watrous have shown that QIP is contained in PSPACE solving a decade-old open problem. The Pontiff has already blessed this result but as someone who has studied both quantum and interactive proofs, let me give you my take. Steve Fenner gives a mostly non-technical outline of the proof below.

QIP stands for Quantum interactive proofs. That PSPACE sits in QIP follows from IP=PSPACE and the fact that quantum interactive proofs can easily simulate classical ones. QIP=PSPACE implies IP=QIP so classical interactive proofs can also simulate quantum ones. Amazing.

Old papers by Watrous and Kitaev and Watrous, and showed two already amazing results:

   1. QIP can be simulated by 3-round quantum interactive proofs.
   2. 3-round quantum interactive proofs can be simulated in exponential time.

The second result proven by creating an exponential-sized semi-definite program to capture the acceptance probability of these proof systems. 

In classical interactive proofs, if we could simulate unbounded rounds with bounded rounds than PSPACE would collapse through the polynomial-time hierarchy (highly unlikely). So while we now know classical interactive proofs can simulate quantum proofs they will need many more rounds to do it.

In an upcoming FOCS paper, Jain, Upadhyay and Watrous show that two-round quantum interactive proof systems can be simulated in PSPACE. QIP in PSPACE uses techniques from all these papers.

Still open: The power of multi-prover quantum interactive proofs where the provers share prior entanglement but otherwise can't communicate. Classically we have MIP=NEXP (nondeterministic exponential time). Does QMIP=MIP? Neither containment is known.
