Related papers: Low-depth arithmetic circuit lower bounds via shif…
Recently, Balliu, Brandt, and Olivetti [FOCS '20] showed the first $\omega(\log^* n)$ lower bound for the maximal independent set (MIS) problem in trees. In this work we prove lower bounds for a much more relaxed family of distributed…
In this work, we prove the strongest known lower bounds for QAC$^0$, allowing polynomially many gates and ancillae. Our main results show that: (1) Depth-3 QAC$^0$ circuits cannot compute PARITY, and require $\Omega(\exp(\sqrt{n}))$ gates…
The paper addresses parametric inequality systems described by polynomial functions in finite dimensions, where state-dependent infinite parameter sets are given by finitely many polynomial inequalities and equalities. Such systems can be…
We study the size blow-up that is necessary to convert an algebraic circuit of product-depth $\Delta+1$ to one of product-depth $\Delta$ in the multilinear setting. We show that for every positive $\Delta = \Delta(n) = o(\log n/\log \log…
Proving super-polynomial size lower bounds for syntactic multilinear Algebraic Branching Programs(smABPs) computing an explicit polynomial is a challenging problem in Algebraic Complexity Theory. The order in which variables in…
We develop tools to study arithmetically induced singular continuous spectrum in the neighborhood of the arithmetic transition in the hyperbolic regime. This leads to first transition-capturing upper bounds on packing and multifractal…
Lower bounds against strong algebraic proof systems and specifically fragments of the Ideal Proof System (IPS), have been obtained in an ongoing line of work. All of these bounds, however, are proved only over large (or characteristic $0$)…
Elementary symmetric polynomials $S_n^k$ are used as a benchmark for the bounded-depth arithmetic circuit model of computation. In this work we prove that $S_n^k$ modulo composite numbers $m=p_1p_2$ can be computed with much fewer…
Low depth measurement-based quantum computation with qudits ($d$-level systems) is investigated and a precise relationship between this powerful model and qudit quantum circuits is derived in terms of computational depth and size…
We show that over the field of complex numbers, \emph{every} homogeneous polynomial of degree $d$ can be approximated (in the border complexity sense) by a depth-$3$ arithmetic circuit of top fan-in at most $d+1$. This is quite surprising…
We study the problem of constructing explicit families of matrices which cannot be expressed as a product of a few sparse matrices. In addition to being a natural mathematical question on its own, this problem appears in various…
We study the fundamental challenge of exhibiting explicit functions that have small correlation with low-degree polynomials over $\mathbb{F}_{2}$. Our main contributions include: 1. In STOC 2020, CHHLZ introduced a new technique to prove…
Valiant's conjecture asserts that the circuit complexity classes VP and VNP are distinct, meaning that the permanent does not admit polynomial-size algebraic circuits. As it is the case in many branches of complexity theory, the…
We numerically investigate the statement that local random quantum circuits acting on n qubits composed of polynomially many nearest neighbour two-qubit gates form an approximate unitary poly(n)-design [F.G.S.L. Brandao et al.,…
There has been a lot of interest recently in proving lower bounds on the size of linear programs needed to represent a given polytope P. In a breakthrough paper Fiorini et al. [Proceedings of 44th ACM Symposium on Theory of Computing 2012,…
A new construction is proposed for low density parity check (LDPC) codes using quadratic permutation polynomials over finite integer rings. The associated graphs for the new codes have both algebraic and pseudo-random nature, and the new…
Obtaining a non-trivial (super-linear) lower bound for computation of the Fourier transform in the linear circuit model has been a long standing open problem for over 40 years. An early result by Morgenstern from 1973, provides an $\Omega(n…
We prove the first unconditional consistency result for superpolynomial circuit lower bounds with a relatively strong theory of bounded arithmetic. Namely, we show that the theory V$^0_2$ is consistent with the conjecture that NEXP…
The Stabbing Planes proof system was introduced to model the reasoning carried out in practical mixed integer programming solvers. As a proof system, it is powerful enough to simulate Cutting Planes and to refute the Tseitin formulas --…
In this article we combine two developments in polynomial optimization. On the one hand, we consider nonnegativity certificates based on sums of nonnegative circuit polynomials, which were recently introduced by the second and the third…