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We generalize some previous results on random polynomials in several complex variables. A standard setting is to consider random polynomials $H_n(z):=\sum_{j=1}^{m_n} a_jp_j(z)$ that are linear combinations of basis polynomials $\{p_j\}$…

In a recent preprint by Deutsch et al. [1995] the authors suggest the possibility of polynomial approximability of arbitrary unitary operations on $n$ qubits by 2-qubit unitary operations. We address that comment by proving strong lower…

By now it is well established that the quantum dimensions of descendants of the adjoint representation can be described in a universal form, independent of a particular family of simple Lie algebras. The Rosso-Jones formula then implies a…

We consider the problem of approximating certain combinatorial polynomials. First, we consider the problem of approximating the Tutte polynomial of a binary matroid with parameters q>= 2 and gamma. (Relative to the classical (x,y)…

We analyse the complexity of computing class polynomials, that are an important ingredient for CM constructions of elliptic curves, via complex floating point approximations of their roots. The heart of the algorithm is the evaluation of…

A surprising 'converse to the polynomial method' of Aaronson et al. (CCC'16) shows that any bounded quadratic polynomial can be computed exactly in expectation by a 1-query algorithm up to a universal multiplicative factor related to the…

We seek random versions of some classical theorems on complex approximation by polynomials and rational functions, as well as investigate properties of random compact sets in connection to complex approximation.

The colored Jones polynomial is a knot invariant that plays a central role in low dimensional topology. We give a simple and an efficient algorithm to compute the colored Jones polynomial of any knot. Our algorithm utilizes the walks along…

Asymptotic approximations for the continuous Hahn polynomials and their zeros as the degree grows to infinity are established via their three-term recurrence relation. The methods are based on the uniform asymptotic expansions for…

We consider a learning-augmented framework for NP-hard permutation problems. The algorithm has access to predictions telling, given a pair $u,v$ of elements, whether $u$ is before $v$ or not in an optimal solution. Building on the work of…

We give simple deterministic reductions demonstrating the NP-hardness of approximating the nearest codeword problem and minimum distance problem within arbitrary constant factors (and almost-polynomial factors assuming NP cannot be solved…

We investigate quantum computational complexity of calculating partition functions of Ising models. We construct a quantum algorithm for an additive approximation of Ising partition functions on square lattices. To this end, we utilize the…

We study algorithms for approximating the permanent of a random matrix when the entries are slightly biased away from zero. This question is motivated by the goal of understanding the classical complexity of linear optics and \emph{boson…

Harrow, Hassidim, and Lloyd showed that for a suitably specified $N \times N$ matrix $A$ and $N$-dimensional vector $\vec{b}$, there is a quantum algorithm that outputs a quantum state proportional to the solution of the linear system of…

We unconditionally prove that it is NP-hard to compute a constant multiplicative approximation to the QUANTUM MAX-CUT problem on an unweighted graph of constant bounded degree. The proof works in two stages: first we demonstrate a generic…

We consider the class of counting problems,i.e. functions in $\#$P, which are self reducible, and have easy decision version, i.e. for every input it is easy to decide if the value of the function $f(x)$ is zero. For example,…

Lawson's iteration is a classical and effective method for solving the linear (polynomial) minimax approximation problem in the complex plane. Extension of Lawson's iteration for the rational minimax approximation problem with both…

We present a randomized algorithm for solving low-degree polynomial equation systems over finite fields faster than exhaustive search. In order to do so, we follow a line of work by Lokshtanov, Paturi, Tamaki, Williams, and Yu (SODA 2017),…

We give a polynomial time approximation scheme (PTAS) for computing the supremum of a Gaussian process. That is, given a finite set of vectors $V\subseteq\mathbb{R}^d$, we compute a $(1+\varepsilon)$-factor approximation to $\mathop…

We prove matching direct and inverse theorems for (algebraic) polynomial approximation with doubling weights $w$ having finitely many zeros and singularities (i.e., points where $w$ becomes infinite) on an interval and not too ``rapidly…