Related papers: Quantum Search for Zeros of Polynomials
We introduce a procedure based on quantum expectation values of measurement observables to characterize quantum coherence. Our measure allows one to quantify coherence without having to perform tomography of the quantum state, and can be…
In this paper, we obtain several new classes of irreducible polynomials having integer coefficients whose zeros lie inside an open disk around the origin or outside a closed annular region in the complex plane. Such irreducible polynomials…
Experimental determination of an unknown quantum state usually requires several incompatible measurements. However, it is also possible to determine the full quantum state from a single, repeated measurement. For this purpose, the quantum…
Quantum computation has attracted much attention since it was shown by Shor and Grover the possibility to implement quantum algorithms able to realize, respectively, factoring and searching in a faster way than any other known classical…
A joint measurement of two observables is a {\it simultaneous} measurement of both quantities upon the {\it same} quantum system. When two quantum-mechanical observables do not commute, then a joint measurement of these observables cannot…
In a recent paper (Cucker, Krick, Malajovich and Wschebor, A Numerical Algorithm for Zero Counting. I: Complexity and accuracy, J. Compl.,24:582-605, 2008) we analyzed a numerical algorithm for computing the number of real zeros of a…
Quantum mechanics can speed up a range of search applications over unsorted data. For example imagine a phone directory containing N names arranged in completely random order. To find someone's phone number with a probability of 50%, any…
In this paper we give a polynomial-time quantum algorithm for computing orders of solvable groups. Several other problems, such as testing membership in solvable groups, testing equality of subgroups in a given solvable group, and testing…
We propose an algorithm based on Newton's method and subdivision for finding all zeros of a polynomial system in a bounded region of the plane. This algorithm can be used to find the intersections between a line and a surface, which has…
The experimental evaluation of many quantum mechanical quantities requires the estimation of several directly measurable observables, such as local observables. Due to the necessity to repeat experiments on individual quantum systems in…
The discovery of derivatives and integrals was a tremendous leap in scientific knowledge and completely revolutionized many fields, including mathematics, physics, and engineering. The existence of higher-order derivatives means better…
Quantum dynamics can be analyzed via the structure of energy eigenstates. However, in the many-body setting, preparing eigenstates associated with finite temperatures requires time scaling exponentially with system size. In this work we…
We consider numerical certification of approximate solutions to a system of polynomial equations with more equations than unknowns by first certifying solutions to a square subsystem. We give several approaches that certifiably select which…
In this contribution, the behavior of zeros of orthogonal polynomials associated with canonical linear spectral transformations of measures supported in the real line is analyzed. An electrostatic interpretation of them is given.
The purpose of this paper is to investigate some properties of q-Euler numbers and polynomials with weight 0. From those q-Euler numbers with weight 0, we derive some identities on the q-Euler numbers and polynomials with weight 0.
We consider sparse polynomials in $N$ variables over a finite field, and ask whether they vanish on a set $S^N$, where $S$ is a set of nonzero elements of the field. We see that if for a polynomial $f$, there is $\mathbf{c}\in S^N$ with $f…
This paper shows that a quantum mechanical algorithm that can query information relating to multiple items of the database, can search a database in a single query (a query is defined as any question to the database to which the database…
Determining the solvability of a given quantum mechanical system is generally challenging. We discuss that the numerical bootstrap method can help us to solve this question in one-dimensional quantum mechanics. We show that the bootstrap…
A simple criterion is derived in order that a number sequence ${\cal S}_n$ is a permitted spectrum of a quantized system. The sequence of the prime numbers fulfils the criterion and the corresponding one-dimensional quantum potential is…
We present a new algorithm for solving the real roots of a bivariate polynomial system $\Sigma=\{f(x,y),g(x,y)\}$ with a finite number of solutions by using a zero-matching method. The method is based on a lower bound for bivariate…