Related papers: A Software Package to Construct Polynomial Sets ov…

We use one photon to simulate an n-qubit quantum system for the first time. We propose a new scheme to realize universal quantum computation in polynomial time O(n^5). A generating set of gates can be realized with high accuracy in the lab.…

Ensuring software correctness remains a fundamental challenge in formal program verification. One promising approach relies on finding polynomial invariants for loops. Polynomial invariants are properties of a program loop that hold before…

We propose a method of compiling that permits to identify quantum circuits able to simulate arbitrary $n$-qubit unitary operations via the adjustment of angles in single-qubit gates therein. The method of compiling itself extends older…

A new method for compiling quantum algorithms is proposed and tested for a three qubit system. The proposed method is to decompose a a unitary matrix U, into a product of simpler U j via a neural network. These U j can then be decomposed…

It is a classical problem to compute a minimal set of invariant polynomial generating the invariant ring of a finite group as an algebra. We present here an algorithm for the computation of minimal generating sets in the non-modular case.…

We develop the theory of Gr\"obner bases for ideals in a polynomial ring with countably infinite variables over a field. As an application we reconstruct some of the one-one correspondences among various sets of partitions by using division…

We present a new open source C library \texttt{msolve} dedicated to solving multivariate polynomial systems of dimension zero through computer algebra methods. The core algorithmic framework of \texttt{msolve} relies on Gr\''obner bases and…

We introduce an open source software package UniversalQCompiler written in Mathematica that allows the decomposition of arbitrary quantum operations into a sequence of single-qubit rotations (with arbitrary rotation angles) and…

Program analysis requires the generation of program properties expressing conditions to hold at intermediate program locations. When it comes to programs with loops, these properties are typically expressed as loop invariants. In this paper…

In this paper, we study the computational complexity of the commutative determinant polynomial computed by a class of set-multilinear circuits which we call regular set-multilinear circuits. Regular set-multilinear circuits are commutative…

The security of multivariate cryptosystems and digital signature schemes relies on the hardness of solving a system of polynomial equations over a finite field. Polynomial system solving is also currently a bottleneck of index-calculus…

We describe how one may go about performing quantum computation with arbitrary "quantum stuff", as long as it has some basic physical properties. Imagine a long strip of stuff, equipped with regularly spaced wires to provide input settings…

In this work, we develop a novel mathematical framework for universal digital quantum computation using algebraic probability theory. We rigorously define quantum circuits as finite sequences of elementary quantum gates and establish their…

We consider cubic polynomials f(z)=z^3+az+b defined over the function field C(L), with a marked point of period N and multiplier L. In the case N=1, there are infinitely many such objects, and in the case N>2, only finitely many. The case…

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…

Polynomial system solving is a classical problem in mathematics with a wide range of applications. This makes its complexity a fundamental problem in computer science. Depending on the context, solving has different meanings. In order to…

Quantum computer provides new opportunities for quantum chemistry. In this article, we present a versatile, extensible, and efficient software package, named Q$^2$Chemistry, for developing quantum algorithms and quantum inspired classical…

It is becoming increasingly clear that, if a useful device for quantum computation will ever be built, it will be embodied by a classical computing machine with control over a truly quantum subsystem, this apparatus performing a mixture of…

Gr\"obner bases can be used for computing the Hilbert basis of a numerical submonoid. By using these techniques, we provide an algorithm that calculates a basis of a subspace of a finite-dimensional vector space over a finite prime field…

We investigate the question if quantum algorithms exist that compute the maximum of a set of conjugated elements of a given number field in quantum polynomial time. We will relate the existence of these algorithms for a certain family of…