Related papers: Improved bounds for reduction to depth 4 and depth…

In this paper we associate to each multivariate polynomial $f$ that is homogeneous relative to a subset of its variables a series of polynomial families $P_\lambda (f)$ of $m$-tuples of homogeneous polynomials of equal degree such that the…

We investigate the complexity of uniform OR circuits and AND circuits of polynomial-size and depth. As their name suggests, OR circuits have OR gates as their computation gates, as well as the usual input, output and constant (0/1) gates.…

We show that for any connected graph $G$ with maximum degree $d\ge3$, the spectral gap from $0$ with respect to the adjacency matrix is at most $\sqrt{d-1}$, with equality if and only if $G$ is the incidence graph of a finite projective…

We consider the quantum query complexity of local search as a function of graph geometry. Given a graph $G = (V,E)$ with $n$ vertices and black box access to a function $f : V \to \mathbb{R}$, the goal is find a vertex $v$ that is a local…

Let $C$ be a depth-3 arithmetic circuit of size at most $s$, computing a polynomial $ f \in \mathbb{F}[x_1,\ldots, x_n] $ (where $\mathbb{F}$ = $\mathbb{Q}$ or $\mathbb{C}$) and the fan-in of the product gates of $C$ is bounded by $d$. We…

Kalai and Kleitman established the bound $n^{\log(d) + 2}$ for the diameter of a $d$-dimensional polyhedron with $n$ facets. Here we improve the bound slightly to $(n-d)^{\log(d)}$.

An $n$-qubit quantum circuit is said to be peaked if it has an output probability that is at least inverse-polynomially large as a function of $n$. We describe a classical algorithm with quasipolynomial runtime $n^{O(\log{n})}$ that…

In this work we establish lower bounds on the size of Clifford circuits that measure a family of commuting Pauli operators. Our bounds depend on the interplay between a pair of graphs: the Tanner graph of the set of measured Pauli…

Reconstruction of arithmetic circuits has been heavily studied in the past few years and has connections to proving lower bounds and deterministic identity testing. In this paper we present a polynomial time randomized algorithm for…

We prove a strongly polynomial bound on the circuit diameter of polyhedra, resolving the circuit analogue of the polynomial Hirsch conjecture. Specifically, we show that the circuit diameter of a polyhedron $P = \{x\in \mathbb{R}^n:\, A x =…

The paper proposes an implicit (i.e., machine-independent) complexity approach to studying computation by polynomial-size, constant-depth circuits with gates counting modulo a constant through the lens of discrete ordinary differential…

We prove lower bounds on complexity measures, such as the approximate degree of a Boolean function and the approximate rank of a Boolean matrix, using quantum arguments. We prove these lower bounds using a quantum query algorithm for the…

This text is a development of a preprint of Ankit Gupta. We present an approach for devising a deterministic polynomial time blackbox identity testing (PIT) algorithm for depth-$4$ circuits with bounded top fanin. This approach is similar…

Let $k$ and $n$ be positive integers, $n>k$. Define $r(n,k)$ to be the minimum positive value of $$ |\sqrt{a_1} + ... + \sqrt{a_k} - \sqrt{b_1} - >... -\sqrt{b_k} | $$ where $ a_1, a_2, ..., a_k, b_1, b_2, ..., b_k $ are positive integers…

Kupavskii proved a codegree version of the Erd\H{o}s--Ko--Rado theorem by showing that for an intersecting family $\mathcal{F} \subseteq \binom{[n]}{k}$ with $n \geq 2k +3d/(1-d/k)$, the minimum $d$-degree of $\mathcal{F}$ is at most…

This paper ties the line of work on algorithms that find an O(sqrt(log(n)))-approximation to the sparsest cut together with the line of work on algorithms that run in sub-quadratic time by using only single-commodity flows. We present an…

We establish an explicit link between depth-3 formulas and one-sided approximation by depth-2 formulas, which were previously studied independently. Specifically, we show that the minimum size of depth-3 formulas is (up to a factor of n)…

We introduce the following variant of the VC-dimension. Given $S \subseteq \{0, 1\}^n$ and a positive integer $d$, we define $\mathbb{U}_d(S)$ to be the size of the largest subset $I \subseteq [n]$ such that the projection of $S$ on every…

We contribute a 2D nearest-neighbor quantum architecture for Shor's algorithm to factor an $n$-bit number in $O(\log^2(n))$ depth. Our implementation uses parallel phase estimation, constant-depth fanout and teleportation, and…

In this work, we show that the class of multivariate degree-$d$ polynomials mapping $\{0,1\}^{n}$ to any Abelian group $G$ is locally correctable with $\widetilde{O}_{d}((\log n)^{d})$ queries for up to a fraction of errors approaching half…