72 research outputs found
Approximate Degree, Secret Sharing, and Concentration Phenomena
The epsilon-approximate degree deg~_epsilon(f) of a Boolean function f is the least degree of a real-valued polynomial that approximates f pointwise to within epsilon. A sound and complete certificate for approximate degree being at least k is a pair of probability distributions, also known as a dual polynomial, that are perfectly k-wise indistinguishable, but are distinguishable by f with advantage 1 - epsilon. Our contributions are:
- We give a simple, explicit new construction of a dual polynomial for the AND function on n bits, certifying that its epsilon-approximate degree is Omega (sqrt{n log 1/epsilon}). This construction is the first to extend to the notion of weighted degree, and yields the first explicit certificate that the 1/3-approximate degree of any (possibly unbalanced) read-once DNF is Omega(sqrt{n}). It draws a novel connection between the approximate degree of AND and anti-concentration of the Binomial distribution.
- We show that any pair of symmetric distributions on n-bit strings that are perfectly k-wise indistinguishable are also statistically K-wise indistinguishable with at most K^{3/2} * exp (-Omega (k^2/K)) error for all k < K <= n/64. This bound is essentially tight, and implies that any symmetric function f is a reconstruction function with constant advantage for a ramp secret sharing scheme that is secure against size-K coalitions with statistical error K^{3/2} * exp (-Omega (deg~_{1/3}(f)^2/K)) for all values of K up to n/64 simultaneously. Previous secret sharing schemes required that K be determined in advance, and only worked for f=AND. Our analysis draws another new connection between approximate degree and concentration phenomena.
As a corollary of this result, we show that for any d deg~_{1/3}(f). These upper and lower bounds were also previously only known in the case f=AND
Δ-Coherent pairs and orthogonal polynomials of a discrete variable
27 pages, no figures.-- MSC1991 codes: 33C25; 42C05.MR#: MR1949214 (2003m:33009)Zbl#: Zbl 1047.42019In this paper we define the concept of D-coherent pair of linear functionals. We prove that if (u_0, u_1) is a D-coherent pair of linear functionals then at least one of them must be a classical discrete linear functional under certain conditions. Examples related to Meixner and Hahn linear functionals are given.F. Marcellán wishes to acknowledge Dirección General de Investigación (MCYT) of Spain
for financial support under grant BFM2000-0206C04-01 and INTAS project 2000-272.Publicad
Constraints on Flavored 2d CFT Partition Functions
We study the implications of modular invariance on 2d CFT partition functions
with abelian or non-abelian currents when chemical potentials for the charges
are turned on, i.e. when the partition functions are "flavored". We begin with
a new proof of the transformation law for the modular transformation of such
partition functions. Then we proceed to apply modular bootstrap techniques to
constrain the spectrum of charged states in the theory. We improve previous
upper bounds on the state with the greatest "mass-to-charge" ratio in such
theories, as well as upper bounds on the weight of the lightest charged state
and the charge of the weakest charged state in the theory. We apply the
extremal functional method to theories that saturate such bounds, and in
several cases we find the resulting prediction for the occupation numbers are
precisely integers. Because such theories sometimes do not saturate a bound on
the full space of states but do saturate a bound in the neutral sector of
states, we find that adding flavor allows the extremal functional method to
solve for some partition functions that would not be accessible to it
otherwise.Comment: 45 pages, 16 Figures v3: typos corrected, expanded appendix on
numeric implementatio
The Conformal Bootstrap: Theory, Numerical Techniques, and Applications
Conformal field theories have been long known to describe the fascinating
universal physics of scale invariant critical points. They describe continuous
phase transitions in fluids, magnets, and numerous other materials, while at
the same time sit at the heart of our modern understanding of quantum field
theory. For decades it has been a dream to study these intricate strongly
coupled theories nonperturbatively using symmetries and other consistency
conditions. This idea, called the conformal bootstrap, saw some successes in
two dimensions but it is only in the last ten years that it has been fully
realized in three, four, and other dimensions of interest. This renaissance has
been possible both due to significant analytical progress in understanding how
to set up the bootstrap equations and the development of numerical techniques
for finding or constraining their solutions. These developments have led to a
number of groundbreaking results, including world record determinations of
critical exponents and correlation function coefficients in the Ising and
models in three dimensions. This article will review these exciting
developments for newcomers to the bootstrap, giving an introduction to
conformal field theories and the theory of conformal blocks, describing
numerical techniques for the bootstrap based on convex optimization, and
summarizing in detail their applications to fixed points in three and four
dimensions with no or minimal supersymmetry.Comment: 81 pages, double column, 58 figures; v3: updated references, minor
typos correcte
On the Fine-Grained Query Complexity of Symmetric Functions
This paper explores a fine-grained version of the Watrous conjecture,
including the randomized and quantum algorithms with success probabilities
arbitrarily close to . Our contributions include the following:
i) An analysis of the optimal success probability of quantum and randomized
query algorithms of two fundamental partial symmetric Boolean functions given a
fixed number of queries. We prove that for any quantum algorithm computing
these two functions using queries, there exist randomized algorithms using
queries that achieve the same success probability as the
quantum algorithm, even if the success probability is arbitrarily close to 1/2.
ii) We establish that for any total symmetric Boolean function , if a
quantum algorithm uses queries to compute with success probability
, then there exists a randomized algorithm using queries to
compute with success probability on a
fraction of inputs, where can be arbitrarily small
positive values. As a corollary, we prove a randomized version of
Aaronson-Ambainis Conjecture for total symmetric Boolean functions in the
regime where the success probability of algorithms can be arbitrarily close to
1/2.
iii) We present polynomial equivalences for several fundamental complexity
measures of partial symmetric Boolean functions. Specifically, we first prove
that for certain partial symmetric Boolean functions, quantum query complexity
is at most quadratic in approximate degree for any error arbitrarily close to
1/2. Next, we show exact quantum query complexity is at most quadratic in
degree. Additionally, we give the tight bounds of several complexity measures,
indicating their polynomial equivalence.Comment: accepted in ISAAC 202
Lightcone expansions of conformal blocks in closed form
We present new closed-form expressions for 4-point scalar conformal blocks in
the s- and t-channel lightcone expansions. Our formulae apply to intermediate
operators of arbitrary spin in general dimensions. For physical spin ,
they are composed of at most Gaussian hypergeometric functions at
each order.Comment: 17 pages, v2, published version, Appendix B added, typos correcte
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