446 research outputs found
Random Weighting, Asymptotic Counting, and Inverse Isoperimetry
For a family X of k-subsets of the set 1,...,n, let |X| be the cardinality of
X and let Gamma(X,mu) be the expected maximum weight of a subset from X when
the weights of 1,...,n are chosen independently at random from a symmetric
probability distribution mu on R. We consider the inverse isoperimetric problem
of finding mu for which Gamma(X,mu) gives the best estimate of ln|X|. We prove
that the optimal choice of mu is the logistic distribution, in which case
Gamma(X,mu) provides an asymptotically tight estimate of ln|X| as k^{-1}ln|X|
grows. Since in many important cases Gamma(X,mu) can be easily computed, we
obtain computationally efficient approximation algorithms for a variety of
counting problems. Given mu, we describe families X of a given cardinality with
the minimum value of Gamma(X,mu), thus extending and sharpening various
isoperimetric inequalities in the Boolean cube.Comment: The revision contains a new isoperimetric theorem, some other
improvements and extensions; 29 pages, 1 figur
Quantum ground state isoperimetric inequalities for the energy spectrum of local Hamiltonians
We investigate the relationship between the energy spectrum of a local
Hamiltonian and the geometric properties of its ground state. By generalizing a
standard framework from the analysis of Markov chains to arbitrary
(non-stoquastic) Hamiltonians we are naturally led to see that the spectral gap
can always be upper bounded by an isoperimetric ratio that depends only on the
ground state probability distribution and the range of the terms in the
Hamiltonian, but not on any other details of the interaction couplings. This
means that for a given probability distribution the inequality constrains the
spectral gap of any local Hamiltonian with this distribution as its ground
state probability distribution in some basis (Eldar and Harrow derived a
similar result in order to characterize the output of low-depth quantum
circuits). Going further, we relate the Hilbert space localization properties
of the ground state to higher energy eigenvalues by showing that the presence
of k strongly localized ground state modes (i.e. clusters of probability, or
subsets with small expansion) in Hilbert space implies the presence of k energy
eigenvalues that are close to the ground state energy. Our results suggest that
quantum adiabatic optimization using local Hamiltonians will inevitably
encounter small spectral gaps when attempting to prepare ground states
corresponding to multi-modal probability distributions with strongly localized
modes, and this problem cannot necessarily be alleviated with the inclusion of
non-stoquastic couplings
Tighter Relations Between Sensitivity and Other Complexity Measures
Sensitivity conjecture is a longstanding and fundamental open problem in the
area of complexity measures of Boolean functions and decision tree complexity.
The conjecture postulates that the maximum sensitivity of a Boolean function is
polynomially related to other major complexity measures. Despite much attention
to the problem and major advances in analysis of Boolean functions in the past
decade, the problem remains wide open with no positive result toward the
conjecture since the work of Kenyon and Kutin from 2004.
In this work, we present new upper bounds for various complexity measures in
terms of sensitivity improving the bounds provided by Kenyon and Kutin.
Specifically, we show that deg(f)^{1-o(1)}=O(2^{s(f)}) and C(f) < 2^{s(f)-1}
s(f); these in turn imply various corollaries regarding the relation between
sensitivity and other complexity measures, such as block sensitivity, via known
results. The gap between sensitivity and other complexity measures remains
exponential but these results are the first improvement for this difficult
problem that has been achieved in a decade.Comment: This is the merged form of arXiv submission 1306.4466 with another
work. Appeared in ICALP 2014, 14 page
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