Related papers: An Algorithm for Constructing Polynomial Systems W…

Quantum algorithms use the principles of quantum mechanics, as for example quantum superposition, in order to solve particular problems outperforming standard computation. They are developed for cryptography, searching, optimisation,…

Among the many proposals for the realization of a quantum computer, holonomic quantum computation (HQC) is distinguished from the rest in that it is geometrical in nature and thus expected to be robust against decoherence. Here we analyze…

Let f be a real or complex polynomial. We give an algorithm to compute the set of generalized critical values. The algorithm uses a finite dimensional space of rational arcs along which we can reach all generalized critical values of f.

When we consider a finite abelian group acting linearly on a polynomial ring, we can find monomial generators for the subring of invariants. By Noether's degree bound and Hilbert's finiteness theorem, we know that there are finitely many…

Many quantum algorithms for attacking symmetric cryptography involve the rank problem of quantum linear equations. In this paper, we first propose two quantum algorithms for solving quantum linear systems of equations with coherent…

Given a quantum algorithm, it is highly nontrivial to devise an efficient sequence of physical gates implementing the algorithm on real hardware and incorporating topological quantum error correction. In this paper, we present a first step…

This paper tackles the problem of constructing Bezout matrices for Newton polynomials in a basis-preserving approach that operates directly with the given Newton basis, thus avoiding the need for transformation from Newton basis to monomial…

This work proposes numerical tests which determine whether a two-qubit operator has an atypically simple quantum circuit. Specifically, we describe formulae, written in terms of matrix coefficients, characterizing operators implementable…

The polynomial method from circuit complexity has been applied to several fundamental problems and obtains the state-of-the-art running times. As observed in [Alman and Williams, STOC 2017], almost all applications of the polynomial method…

In this paper, a polynomial-time algorithm is given to compute the generalized Hermite normal form for a matrix F over Z[x], or equivalently, the reduced Groebner basis of the Z[x]-module generated by the column vectors of F. The algorithm…

We suggest a mathematical definition of the notion of master integrals and present a brief review of algorithmic methods to solve reduction problems for Feynman integrals based on integration by parts relations. In particular, we discuss a…

We show that for any constant d, complex roots of degree d univariate rational (or Gaussian rational) polynomials---given by a list of coefficients in binary---can be computed to a given accuracy by a uniform TC^0 algorithm (a uniform…

The progress in building quantum computers to execute quantum algorithms has recently been remarkable. Grover's search algorithm in a binary quantum system provides considerable speed-up over classical paradigm. Further, Grover's algorithm…

We consider bivariate piecewise polynomial finite element spaces for curved domains bounded by piecewise conics satisfying homogeneous boundary conditions, construct stable local bases for them using Bernstein-B\'ezier techniques, prove…

We propose a method to compute the numerical solutions of a polynomial system in complete intersection. This algorithm makes use of Bezout matrices and need only linear algebra computations. All the calculations can be done in floating…

The image of a polynomial map is a constructible set. While computing its closure is standard in computer algebra systems, a procedure for computing the constructible set itself is not. We provide a new algorithm, based on algebro-geometric…

Let $\K$ be a field and $(f_1, \ldots, f_n)\subset \K[X_1, \ldots, X_n]$ be a sequence of quasi-homogeneous polynomials of respective weighted degrees $(d_1, \ldots, d_n)$ w.r.t a system of weights $(w_{1},\dots,w_{n})$. Such systems are…

Primitive polynomials over finite fields are crucial for various domains of computer science, including classical pseudo-random number generation, coding theory and post-quantum cryptography. Nevertheless, the pursuit of an efficient…

For a native gate set which includes all single-qubit gates, we apply results from symplectic geometry to analyze the spaces of two-qubit programs accessible within a fixed number of gates. These techniques yield an explicit description of…

Computation of (approximate) polynomials common factors is an important problem in several fields of science, like control theory and signal processing. While the problem has been widely studied for scalar polynomials, the scientific…