02256nas a2200121 4500008004100000245005500041210005400096260001500150520188900165100002002054700002302074856003702097 2010 eng d00aPseudorandom generators and the BQP vs. PH problem0 aPseudorandom generators and the BQP vs PH problem c2010/07/023 a It is a longstanding open problem to devise an oracle relative to which BQP
does not lie in the Polynomial-Time Hierarchy (PH). We advance a natural
conjecture about the capacity of the Nisan-Wigderson pseudorandom generator
[NW94] to fool AC_0, with MAJORITY as its hard function. Our conjecture is
essentially that the loss due to the hybrid argument (which is a component of
the standard proof from [NW94]) can be avoided in this setting. This is a
question that has been asked previously in the pseudorandomness literature
[BSW03]. We then make three main contributions: (1) We show that our conjecture
implies the existence of an oracle relative to which BQP is not in the PH. This
entails giving an explicit construction of unitary matrices, realizable by
small quantum circuits, whose row-supports are "nearly-disjoint." (2) We give a
simple framework (generalizing the setting of Aaronson [A10]) in which any
efficiently quantumly computable unitary gives rise to a distribution that can
be distinguished from the uniform distribution by an efficient quantum
algorithm. When applied to the unitaries we construct, this framework yields a
problem that can be solved quantumly, and which forms the basis for the desired
oracle. (3) We prove that Aaronson's "GLN conjecture" [A10] implies our
conjecture; our conjecture is thus formally easier to prove. The GLN conjecture
was recently proved false for depth greater than 2 [A10a], but it remains open
for depth 2. If true, the depth-2 version of either conjecture would imply an
oracle relative to which BQP is not in AM, which is itself an outstanding open
problem. Taken together, our results have the following interesting
interpretation: they give an instantiation of the Nisan-Wigderson generator
that can be broken by quantum computers, but not by the relevant modes of
classical computation, if our conjecture is true.
1 aFefferman, Bill1 aUmans, Christopher uhttp://arxiv.org/abs/1007.0305v3