In earlier work, we gave an oracle separating the relational versions of BQP and the polynomial hierarchy, and showed that an oracle separating the decision versions would follow from what we called the Generalized Linial-Nisan (GLN) Conjecture: that "almost k-wise independent" distributions are indistinguishable from the uniform distribution by constant-depth circuits. The original Linial-Nisan Conjecture was recently proved by Braverman; we offered a $200 prize for the generalized version. In this paper, we save ourselves $200 by showing that the GLN Conjecture is false, at least for circuits of depth 3 and higher. As a byproduct, our counterexample also implies that Pi2P is not contained in P^NP relative to a random oracle with probability 1. It has been conjectured since the 1980s that PH is infinite relative to a random oracle, but the highest levels of PH previously proved separate were NP and coNP. Finally, our counterexample implies that the famous results of Linial, Mansour, and Nisan, on the structure of AC0 functions, cannot be improved in several interesting respects.
Proving an oracle separation between BQP and PH is one of the central open problems of quantum complexity theory. In a recent paper [1], we reported the following progress on the problem: (1) We constructed an oracle relative to which FBQP ⊂ FBPP PH , where FBQP and FBPP PH are the "relational" versions of BQP and PH respectively (that is, the versions where there are many valid outputs, and an algorithm's task is to output any one of them).
(2) We proposed a natural decision problem, called Fourier Checking, which is provably in BQP (as an oracle problem) and which we conjectured was not in PH.
(3) We showed that Fourier Checking has a property called almost k-wise independence, and that no BPP path or SZK problem shares that property. This allowed us to give oracles relative to which BQP was outside those classes, and to reprove all the known oracle separations between BQP and classical complexity classes in a unified way.
(4) We conjectured that no PH problem has the almost k-wise independence property, and called that the Generalized Linial-Nisan (GLN) Conjecture. Proving the GLN Conjecture would imply the existence of an oracle relative to which BQP ⊂ PH.
This paper does nothing to modify points (1)-(3) above: the unconditional results in [1] are still true, and we still conjecture not only that there exists an oracle relative to which BQP ⊂ PH, but that Fourier Checking is such an oracle.
However, we will show that the hope of proving Fourier Checking / ∈ PH by proving the GLN Conjecture was unfounded:
The GLN Conjecture is false, at least for Π p 2 and higher levels of the polynomial hierarchy.
We prove this by giving an explicit counterexample: a family of depth-three AC 0 circuits that distinguish the uniform distribution over n-bit strings from an O (k/n)-almost k-wise independent distribution, with constant bias. 1Our counterexample was inspired by a recent result of Beame and Machmouchi [3], giving a Boolean function with quantum query complexity Ω (n/ log n) that is computable by a depth-three AC 0 circuit. This disproved a conjecture, relayed to us earlier by Beame, stating that every AC 0 function has quantum query complexity n 1-Ω (1) . Like the Beame-Machmouchi counterexample, ours involves inputs X = x 1 . . . x N ∈ [M ] N that are lists of positive integers, with the x i ’s encoded in binary to obtain a Boolean problem; as well as a function f : [M ] N → {0, 1} that uses two alternating quantifiers to express a “global” property of X. In Beame and Machmouchi’s case, the property in question was that the function x (i) := x i is 2-to-1; in our case, the property is that x (i) is surjective. 2Our counterexample makes essential use of depth-three circuits, and we find it plausible that the GLN Conjecture still holds for depth-two circuits (i.e., for DNF formulas). 3 As shown in [1], proving the GLN Conjecture for depth-two circuits would yield an oracle relative to which BQP ⊂ AM, which is already a longstanding open problem.
Given that the GLN Conjecture resisted attacks for two years (and indirectly motivated the beautiful works of Razborov [16] and Braverman [7] on the original LN Conjecture), our counterexample cannot have been quite as obvious as it seems in retrospect! Perhaps Andy Drucker (personal communication) summarized the situation best: almost k-wise independent distributions seem to be much better at fooling people than at fooling circuits.
Besides falsifying the GLN Conjecture, our counterexample has several other interesting implications for PH and AC 0 .
Firstly, we are able to use our counterexample to prove that (Π p 2 )
A ⊂ P NP A with probability 1 relative to a random oracle A. Indeed, we conjecture that our counterexample can even be used to prove (Π p 2 )
A ⊂ (Σ p 2 )
A with probability 1 for a random oracle A. The seminal work of Yao [18] showed PH infinite relative to some oracle, but it has been an open problem for almost thirty years to prove PH infinite relative to a random oracle (see the book of Håstad [17] for discussion).
Motivation for this problem comes from a surprising result of Book [6], which says that if PH collapses relative to a random oracle, then it also collapses in the unrelativized world. Our result, while simple, appears to represent the first “progress” toward separating PH by random oracles since the original result of Bennett and Gill [5] that P = NP = coNP relative to a random oracle with probability 1. 4Secondly, our counterexample shows that the celebrated results of Linial, Mansour, and Nisan [12], on the Fourier spectrum of AC 0 functions, cannot be improved in several important respects. In particular, Linial et al. showed that every Boolean function f : {0, 1} n → {0, 1} in AC 0 has average sensitivity O (polylog (n)). However, we observe that this result fails completely if we consider a closely-related measure, the average block-sensitivity. Indeed, there exists a reasonablybalanced Boolean function f ∈ AC 0 such that
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