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We present the first computationally-efficient algorithm for average-case learning of shallow quantum circuits with many-qubit gates. Specifically, we provide a quasi-polynomial time and sample complexity algorithm for learning unknown…
We consider the construction of maximal families of polynomials over the finite field $\mathbb{F}_q$, all having the same degree $n$ and a nonzero constant term, where the degree of the GCD of any two polynomials is $d$ with $1 \le d\le n$.…
General error locator polynomials are polynomials able to decode any correctable syndrome for a given linear code. Such polynomials are known to exist for all cyclic codes and for a large class of linear codes. We provide some decoding…
We study the non-linear extension of integer programming with greatest common divisor constraints of the form $\gcd(f,g) \sim d$, where $f$ and $g$ are linear polynomials, $d$ is a positive integer, and $\sim$ is a relation among $\leq, =,…
A new algorithm to compute cylindrical algebraic decompositions (CADs) is presented, building on two recent advances. Firstly, the output is truth table invariant (a TTICAD) meaning given formulae have constant truth value on each cell of…
An algorithm and associated strategy for solving polynomial systems within the optimization framework is presented. The algorithm and strategy are named, respectively, the penetrating gradient algorithm and the deepest descent strategy. The…
With the development of quantum hardware bringing the error-corrected quantum circuits to the near future, the lack of an efficient polynomial-time decoding algorithms for logical circuits presents a critical bottleneck. While quantum…
The {\sc $c$-Balanced Separator} problem is a graph-partitioning problem in which given a graph $G$, one aims to find a cut of minimum size such that both the sides of the cut have at least $cn$ vertices. In this paper, we present new…
Cylindrical algebraic decomposition (CAD) is an important tool for the study of real algebraic geometry with many applications both within mathematics and elsewhere. It is known to have doubly exponential complexity in the number of…
We propose a symbolic-numeric algorithm to count the number of solutions of a polynomial system within a local region. More specifically, given a zero-dimensional system $f_1=\cdots=f_n=0$, with $f_i\in\mathbb{C}[x_1,\ldots,x_n]$, and a…
In this paper we revisit the greatest common right divisor (GCRD) extraction from a set of polynomial matrices $P_i(\lambda)\in \F[\la]^{m_i\times n}$, $i=1,\ldots,k$ with coefficients in a generic field $\F$, and with common column…
Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that,…
The cyclic block coordinate descent-type (CBCD-type) methods, which performs iterative updates for a few coordinates (a block) simultaneously throughout the procedure, have shown remarkable computational performance for solving strongly…
This work proposes a method for solving linear stochastic optimal control (SOC) problems using sum of squares and semidefinite programming. Previous work had used polynomial optimization to approximate the value function, requiring a high…
Algorithms which compute modulo triangular sets must respect the presence of zero-divisors. We present Hensel lifting as a tool for dealing with them. We give an application: a modular algorithm for computing GCDs of univariate polynomials…
Many problems of systems control theory boil down to solving polynomial equations, polynomial inequalities or polyomial differential equations. Recent advances in convex optimization and real algebraic geometry can be combined to generate…
We present algorithmic, complexity and implementation results concerning real root isolation of integer univariate polynomials using the continued fraction expansion of real algebraic numbers. One motivation is to explain the method's good…
Linear reversible circuits represent a subclass of reversible circuits with many applications in quantum computing. These circuits can be efficiently simulated by classical computers and their size is polynomially bounded by the number of…
We develop efficient randomized algorithms to solve the black-box reconstruction problem for polynomials over finite fields, computable by depth three arithmetic circuits with alternating addition/multiplication gates, such that output gate…
Given two polynomials, we find a convergence property of the GCD of the rising factorial and the falling factorial. Based on this property, we present a unified approach to computing the universal denominators as given by Gosper's algorithm…