69 research outputs found
Gap Amplification for Small-Set Expansion via Random Walks
In this work, we achieve gap amplification for the Small-Set Expansion
problem. Specifically, we show that an instance of the Small-Set Expansion
Problem with completeness and soundness is at least as
difficult as Small-Set Expansion with completeness and soundness
, for any function which grows faster than
. We achieve this amplification via random walks -- our gadget
is the graph with adjacency matrix corresponding to a random walk on the
original graph. An interesting feature of our reduction is that unlike gap
amplification via parallel repetition, the size of the instances (number of
vertices) produced by the reduction remains the same
The Equivalence of Sampling and Searching
In a sampling problem, we are given an input x, and asked to sample
approximately from a probability distribution D_x. In a search problem, we are
given an input x, and asked to find a member of a nonempty set A_x with high
probability. (An example is finding a Nash equilibrium.) In this paper, we use
tools from Kolmogorov complexity and algorithmic information theory to show
that sampling and search problems are essentially equivalent. More precisely,
for any sampling problem S, there exists a search problem R_S such that, if C
is any "reasonable" complexity class, then R_S is in the search version of C if
and only if S is in the sampling version. As one application, we show that
SampP=SampBQP if and only if FBPP=FBQP: in other words, classical computers can
efficiently sample the output distribution of every quantum circuit, if and
only if they can efficiently solve every search problem that quantum computers
can solve. A second application is that, assuming a plausible conjecture, there
exists a search problem R that can be solved using a simple linear-optics
experiment, but that cannot be solved efficiently by a classical computer
unless the polynomial hierarchy collapses. That application will be described
in a forthcoming paper with Alex Arkhipov on the computational complexity of
linear optics.Comment: 16 page
On Fortification of Projection Games
A recent result of Moshkovitz \cite{Moshkovitz14} presented an ingenious
method to provide a completely elementary proof of the Parallel Repetition
Theorem for certain projection games via a construction called fortification.
However, the construction used in \cite{Moshkovitz14} to fortify arbitrary
label cover instances using an arbitrary extractor is insufficient to prove
parallel repetition. In this paper, we provide a fix by using a stronger graph
that we call fortifiers. Fortifiers are graphs that have both and
guarantees on induced distributions from large subsets. We then show
that an expander with sufficient spectral gap, or a bi-regular extractor with
stronger parameters (the latter is also the construction used in an independent
update \cite{Moshkovitz15} of \cite{Moshkovitz14} with an alternate argument),
is a good fortifier. We also show that using a fortifier (in particular
guarantees) is necessary for obtaining the robustness required for
fortification.Comment: 19 page
Parallel Repetition for the GHZ Game: Exponential Decay
We show that the value of the -fold repeated GHZ game is at most
, improving upon the polynomial bound established by Holmgren
and Raz. Our result is established via a reduction to approximate subgroup type
questions from additive combinatorics
Upper Tail Estimates with Combinatorial Proofs
We study generalisations of a simple, combinatorial proof of a Chernoff bound
similar to the one by Impagliazzo and Kabanets (RANDOM, 2010).
In particular, we prove a randomized version of the hitting property of
expander random walks and apply it to obtain a concentration bound for expander
random walks which is essentially optimal for small deviations and a large
number of steps. At the same time, we present a simpler proof that still yields
a "right" bound settling a question asked by Impagliazzo and Kabanets.
Next, we obtain a simple upper tail bound for polynomials with input
variables in which are not necessarily independent, but obey a certain
condition inspired by Impagliazzo and Kabanets. The resulting bound is used by
Holenstein and Sinha (FOCS, 2012) in the proof of a lower bound for the number
of calls in a black-box construction of a pseudorandom generator from a one-way
function.
We then show that the same technique yields the upper tail bound for the
number of copies of a fixed graph in an Erd\H{o}s-R\'enyi random graph,
matching the one given by Janson, Oleszkiewicz and Ruci\'nski (Israel J. Math,
2002).Comment: Full version of the paper from STACS 201
Unbounded violations of bipartite Bell Inequalities via Operator Space theory
In this work we show that bipartite quantum states with local Hilbert space
dimension n can violate a Bell inequality by a factor of order (up
to a logarithmic factor) when observables with n possible outcomes are used. A
central tool in the analysis is a close relation between this problem and
operator space theory and, in particular, the very recent noncommutative
embedding theory. As a consequence of this result, we obtain better Hilbert
space dimension witnesses and quantum violations of Bell inequalities with
better resistance to noise
Communication Complexity of Statistical Distance
We prove nearly matching upper and lower bounds on the randomized communication complexity of the following problem: Alice and Bob are each given a probability distribution over elements, and they wish to estimate within +-epsilon the statistical (total variation) distance between their distributions. For some range of parameters, there is up to a log(n) factor gap between the upper and lower bounds, and we identify a barrier to using information complexity techniques to improve the lower bound in this case. We also prove a side result that we discovered along the way: the randomized communication complexity of n-bit Majority composed with n-bit Greater-Than is Theta(n log n)
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