Related papers: Observations on Symmetric Circuits

Comparator circuits are a natural circuit model for studying bounded fan-out computation whose power sits between nondeterministic branching programs and general circuits. Despite having been studied for nearly three decades, the first…

Exponential sums with monomials are highly related to many interesting problems in number theory and well studied by many literatures. In this paper, we consider the exponential sums with polynomials and prove a new upper bound. As an…

We give new lower and upper bounds on the permanent of a doubly stochastic matrix. Combined with previous work, this improves on the deterministic approximation factor for the permanent. We also give a combinatorial application of the lower…

The method of shifted partial derivatives was used to prove a super-polynomial lower bound on the size of depth four circuits needed to compute the permanent. We show that this method alone cannot prove that the padded permanent $\ell^{n-m}…

The problem of constructing hazard-free Boolean circuits dates back to the 1940s and is an important problem in circuit design. Our main lower-bound result unconditionally shows the existence of functions whose circuit complexity is…

Circuit polynomials are a certificate of nonnegativity for real polynomials, which can be derived via a generalization of the classical inequality of arithmetic and geometric means. In this article, we show that similarly nonnegativity of…

In recent years, there has been a flurry of activity towards proving lower bounds for homogeneous depth-4 arithmetic circuits, which has brought us very close to statements that are known to imply $\textsf{VP} \neq \textsf{VNP}$. It is open…

We prove that $poly(t) \cdot n^{1/D}$-depth local random quantum circuits with two qudit nearest-neighbor gates on a $D$-dimensional lattice with n qudits are approximate $t$-designs in various measures. These include the "monomial"…

We show that the permanent cannot be computed by DLOGTIME-uniform threshold or arithmetic circuits of depth o(log log n) and polynomial size.

We study the power of negation in the Boolean and algebraic settings and show the following results. * We construct a family of polynomials $P_n$ in $n$ variables, all of whose monomials have positive coefficients, such that $P_n$ can be…

When the gate set has continuous parameters, synthesizing a unitary operator as a quantum circuit is always possible using exact methods, but finding minimal circuits efficiently remains a challenging problem. The landscape is very…

Proving that there are problems in $\mathsf{P}^\mathsf{NP}$ that require boolean circuits of super-linear size is a major frontier in complexity theory. While such lower bounds are known for larger complexity classes, existing results only…

We show that deciding simulation equivalence and simulation preorder have quadratic lower bounds assuming that the Strong Exponential Time Hypothesis holds. This is in line with the best know quadratic upper bounds of simulation…

Let $S_n$ denote a symmetric group, $\chi$ an irreducible character of $S_n$, and $g\in S_n$ an element which decomposes into $k$ disjoint cycles, where $1$-cycles are included. Then $|\chi(g)|\le k!$, and this upper bound is sharp for…

$f,g_1,...,g_m$ be elements of the polynomial ring $\mathbb{R}[x_1,...,x_n]$. The paper deals with the general problem of computing a lower bound for $f$ on the subset of $\mathbb{R}^n$ defined by the inequalities $g_i\ge 0$, $i=1,...,m$.…

We give new, explicit and asymptotically sharp, lower bounds for dimensions of irreducible modular representations of finite symmetric groups.

This article is devoted to the study of discrete potentials on the sphere in $\mathbb{R}^n$ for sharp codes. We show that the potentials of most of the known sharp codes attain the universal lower bounds for polarization for spherical…

We introduce a method for proving lower bounds on the efficacy of semidefinite programming (SDP) relaxations for combinatorial problems. In particular, we show that the cut, TSP, and stable set polytopes on $n$-vertex graphs are not the…

For every constant $d$, we design a subexponential time deterministic algorithm that takes as input a multivariate polynomial $f$ given as a constant depth algebraic circuit over the field of rational numbers, and outputs all irreducible…

We define eventually symmetric functions to be those power series of bounded degree in infinitely many variables that are invariant under interchanging all the variables with large enough indices. We show how this ring $\tilde{\Lambda}$ is…