Related papers: A quantum algorithm for approximating the influenc…
This paper explores a fine-grained version of the Watrous conjecture, including the randomized and quantum algorithms with success probabilities arbitrarily close to $1/2$. Our contributions include the following: i) An analysis of the…
Gate quantum computers generate significant interest due to their potential to solve certain difficult problems such as prime factorization in polynomial time. Computer vision researchers have long been attracted to the power of quantum…
Quantum computers promise to redefine the boundaries of computational science, offering the potential for exponential speedups in solving complex problems across chemistry, optimization, and materials science. Yet, their practical utility…
We prove various theorems on approximation using polynomials with integer coefficients in the Bernstein basis of any given order. In the extreme, we draw the coefficients from $\{ \pm 1\}$ only. A basic case of our results states that for…
This paper considers the problem of approximating a Boolean function $f$ using another Boolean function from a specified class. Two classes of approximating functions are considered: $k$-juntas, and linear Boolean functions. The $n$ input…
In this paper, we study the following variant of the junta learning problem. We are given oracle access to a Boolean function $f$ on $n$ variables that only depends on $k$ variables, and, when restricted to them, equals some predefined…
A Boolean function $f:\{0,1\}^n \to \{0,1\}$ is said to be noise sensitive if inserting a small random error in its argument makes the value of the function almost unpredictable. Benjamini, Kalai and Schramm showed that if the sum of…
We propose the first near-optimal quantum algorithm for estimating in Euclidean norm the mean of a vector-valued random variable with finite mean and covariance. Our result aims at extending the theory of multivariate sub-Gaussian…
We give a {\em deterministic} algorithm for approximately computing the fraction of Boolean assignments that satisfy a degree-$2$ polynomial threshold function. Given a degree-2 input polynomial $p(x_1,\dots,x_n)$ and a parameter $\eps >…
Consider a Boolean function f on the n-dimensional hypercube, and a set of variables (indexed by) $S \subset \{1,2,\ldots,n\}.$ The coalition influence of the variables S on a function f is the probability that after a random assignment of…
The Banzhaf power and interaction indexes for a pseudo-Boolean function (or a cooperative game) appear naturally as leading coefficients in the standard least squares approximation of the function by a pseudo-Boolean function of a specified…
Nisan and Szegedy (CC 1994) showed that any Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$ that depends on all its input variables, when represented as a real-valued multivariate polynomial $P(x_1,\ldots,x_n)$, has degree at least $\log…
This paper studies the important problem of quantum classification of Boolean functions from a entirely novel perspective. Typically, quantum classification algorithms allow us to classify functions with a probability of $1.0$, if we are…
We associate to each Boolean function a polynomial whose evaluations represents the distances from all possible Boolean affine functions. Both determining the coefficients of this polynomial from the truth table of the Boolean function and…
In this article we develop quantum algorithms for learning and testing juntas, i.e. Boolean functions which depend only on an unknown set of k out of n input variables. Our aim is to develop efficient algorithms: - whose sample complexity…
We prove the following conjecture, raised by Aaronson and Ambainis in 2008: Let $f:\{-1,1\}^n \rightarrow [-1,1]$ be a multilinear polynomial of degree $d$. Then there exists a variable $x_i$ whose influence on $f$ is at least…
Let $\mathcal{F}_{n}^*$ be the set of Boolean functions depending on all $n$ variables. We prove that for any $f\in \mathcal{F}_{n}^*$, $f|_{x_i=0}$ or $f|_{x_i=1}$ depends on the remaining $n-1$ variables, for some variable $x_i$. This…
A statistical, data-driven method is presented that quantifies influences between variables of a dynamical system. The method is based on finding a suitable representation of points by fuzzy affiliations with respect to landmark points…
Here we revisit the quantum algorithms for obtaining Forrelation [Aaronson et al, 2015] values to evaluate some of the well-known cryptographically significant spectra of Boolean functions, namely the Walsh spectrum, the cross-correlation…
We present a regularity lemma for Boolean functions $f:\{-1,1\}^n \to \{-1,1\}$ based on noisy influence, a measure of how locally correlated $f$ is with each input bit. We provide an application of the regularity lemma to weaken the…