BQP vs QMA is a Quantum analogy of P vs NP.

I recently went through the below pre-print on the IACR (International Association for Cryptologic Research) webpage.

$BQP\neq QMA$ [link]

I find it rather surprising as this would imply resolving several conjectures such as $P\neq PP$ and $P\neq PSPACE$.

Reason: As $P\subseteq BQP$ (trivial consequence due to their definition), $QMA\subseteq PP$ (due to Kitaev-Waltrous, 2005) and $QMA \subseteq PSPACE$.

Has someone reviewed the paper? How plausible is their argument?


1 Answer 1


I haven't looked at the paper carefully, but one thing I noticed is that their proof that BQP $\subsetneqq$ QMA works by their claiming that "bit commitment $\not \in$ BQP" but "bit commitment $\in$ QMA". However, bit commitment is not a decision problem but a protocol, and BQP and QMA are classes of decision problems.

So unless you can define new complexity classes BQP$^*$ and QMA$^*$ to encompass protocols, and show that BQP$^*$ $\neq$ QMA$^*$ implies BQP $\neq$ QMA (which this paper doesn't do), I don't see how you can derive any conclusions about the standard complexity classes BQP and QMA from this paper.

The fact that bit commitment cannot be done by a quantum protocol is a well-known result. Note that I have not checked their proof of bit commitment $\in$ QMA, in part because I don't understand how they have implicitly expanded QMA to encompass protocols.


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