46 research outputs found
A Counterexample to the Generalized Linial-Nisan Conjecture
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
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.Comment: 17 page
Pseudorandom generators and the BQP vs. PH problem
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.Comment: Updated in light of counterexample to the GLN conjectur
An average-case depth hierarchy theorem for Boolean circuits
We prove an average-case depth hierarchy theorem for Boolean circuits over
the standard basis of , , and gates.
Our hierarchy theorem says that for every , there is an explicit
-variable Boolean function , computed by a linear-size depth- formula,
which is such that any depth- circuit that agrees with on fraction of all inputs must have size This
answers an open question posed by H{\aa}stad in his Ph.D. thesis.
Our average-case depth hierarchy theorem implies that the polynomial
hierarchy is infinite relative to a random oracle with probability 1,
confirming a conjecture of H{\aa}stad, Cai, and Babai. We also use our result
to show that there is no "approximate converse" to the results of Linial,
Mansour, Nisan and Boppana on the total influence of small-depth circuits, thus
answering a question posed by O'Donnell, Kalai, and Hatami.
A key ingredient in our proof is a notion of \emph{random projections} which
generalize random restrictions
Three Puzzles on Mathematics, Computation, and Games
In this lecture I will talk about three mathematical puzzles involving
mathematics and computation that have preoccupied me over the years. The first
puzzle is to understand the amazing success of the simplex algorithm for linear
programming. The second puzzle is about errors made when votes are counted
during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure
Sum of Us: Strategyproof Selection from the Selectors
We consider directed graphs over a set of n agents, where an edge (i,j) is
taken to mean that agent i supports or trusts agent j. Given such a graph and
an integer k\leq n, we wish to select a subset of k agents that maximizes the
sum of indegrees, i.e., a subset of k most popular or most trusted agents. At
the same time we assume that each individual agent is only interested in being
selected, and may misreport its outgoing edges to this end. This problem
formulation captures realistic scenarios where agents choose among themselves,
which can be found in the context of Internet search, social networks like
Twitter, or reputation systems like Epinions.
Our goal is to design mechanisms without payments that map each graph to a
k-subset of agents to be selected and satisfy the following two constraints:
strategyproofness, i.e., agents cannot benefit from misreporting their outgoing
edges, and approximate optimality, i.e., the sum of indegrees of the selected
subset of agents is always close to optimal. Our first main result is a
surprising impossibility: for k \in {1,...,n-1}, no deterministic strategyproof
mechanism can provide a finite approximation ratio. Our second main result is a
randomized strategyproof mechanism with an approximation ratio that is bounded
from above by four for any value of k, and approaches one as k grows
Improved Pseudorandom Generators from Pseudorandom Multi-Switching Lemmas
We give the best known pseudorandom generators for two touchstone classes in
unconditional derandomization: an -PRG for the class of size-
depth- circuits with seed length , and an -PRG for the class of -sparse
polynomials with seed length . These results bring the state of the art for
unconditional derandomization of these classes into sharp alignment with the
state of the art for computational hardness for all parameter settings:
improving on the seed lengths of either PRG would require breakthrough progress
on longstanding and notorious circuit lower bounds.
The key enabling ingredient in our approach is a new \emph{pseudorandom
multi-switching lemma}. We derandomize recently-developed
\emph{multi}-switching lemmas, which are powerful generalizations of
H{\aa}stad's switching lemma that deal with \emph{families} of depth-two
circuits. Our pseudorandom multi-switching lemma---a randomness-efficient
algorithm for sampling restrictions that simultaneously simplify all circuits
in a family---achieves the parameters obtained by the (full randomness)
multi-switching lemmas of Impagliazzo, Matthews, and Paturi [IMP12] and
H{\aa}stad [H{\aa}s14]. This optimality of our derandomization translates into
the optimality (given current circuit lower bounds) of our PRGs for
and sparse polynomials
Exponential Quantum Speed-ups are Generic
A central problem in quantum computation is to understand which quantum
circuits are useful for exponential speed-ups over classical computation. We
address this question in the setting of query complexity and show that for
almost any sufficiently long quantum circuit one can construct a black-box
problem which is solved by the circuit with a constant number of quantum
queries, but which requires exponentially many classical queries, even if the
classical machine has the ability to postselect.
We prove the result in two steps. In the first, we show that almost any
element of an approximate unitary 3-design is useful to solve a certain
black-box problem efficiently. The problem is based on a recent oracle
construction of Aaronson and gives an exponential separation between quantum
and classical bounded-error with postselection query complexities.
In the second step, which may be of independent interest, we prove that
linear-sized random quantum circuits give an approximate unitary 3-design. The
key ingredient in the proof is a technique from quantum many-body theory to
lower bound the spectral gap of local quantum Hamiltonians.Comment: 24 pages. v2 minor correction
The Power of Quantum Fourier Sampling
A line of work initiated by Terhal and DiVincenzo and Bremner, Jozsa, and
Shepherd, shows that quantum computers can efficiently sample from probability
distributions that cannot be exactly sampled efficiently on a classical
computer, unless the PH collapses. Aaronson and Arkhipov take this further by
considering a distribution that can be sampled efficiently by linear optical
quantum computation, that under two feasible conjectures, cannot even be
approximately sampled classically within bounded total variation distance,
unless the PH collapses.
In this work we use Quantum Fourier Sampling to construct a class of
distributions that can be sampled by a quantum computer. We then argue that
these distributions cannot be approximately sampled classically, unless the PH
collapses, under variants of the Aaronson and Arkhipov conjectures.
In particular, we show a general class of quantumly sampleable distributions
each of which is based on an "Efficiently Specifiable" polynomial, for which a
classical approximate sampler implies an average-case approximation. This class
of polynomials contains the Permanent but also includes, for example, the
Hamiltonian Cycle polynomial, and many other familiar #P-hard polynomials.
Although our construction, unlike that proposed by Aaronson and Arkhipov,
likely requires a universal quantum computer, we are able to use this
additional power to weaken the conjectures needed to prove approximate sampling
hardness results