Related papers: Quantum Search for Zeros of Polynomials
In this paper we study the complexity of quantum query algorithms computing the value of Boolean function and its relation to the degree of algebraic polynomial representing this function. We pay special attention to Boolean functions with…
Some purely chiral fractional quantum Hall states are described by symmetric or anti-symmetric polynomials of infinite variables. In this article, we review a systematic construction and classification of those fractional quantum Hall…
In this paper, we derive new bounds for the zeros of quaternionic polynomials by applying localization theorems, which includes Gershgorin-type theorems for the left eigenvalues of matrices of left monic quaternionic polynomials. These…
We establish how the coefficients of a sparse polynomial system influence the sum (or the trace) of its zeros. As an application, we develop numerical tests for verifying whether a set of solutions to a sparse system is complete. These…
The approximate numerical method for a calculation of a quantum wave impedance in a case of a potential energy with a complicated spatial structure is considered. It was proved that the approximation of a real potential by a piesewise…
This paper introduces the notion of probabilistic zero bounds for random polynomials. It presents new results regarding the probabilistic bounds of random polynomials whose coefficients are independently and identically distributed as…
To measure an observable of a quantum mechanical system leaves it in one of its eigenstates and the result of the measurement is one of its eigenvalues. This process is shown to be a computational resource. It allows one, in principle, to…
We present a zero decomposition theorem and an algorithm based on Wu's method, which computes a zero decomposition with multiplicity for a given zero-dimensional polynomial system. If the system satisfies some condition, the zero…
The concept of intrinsic and operational observables in quantum mechanics is introduced. It is argued that, in any realistic description of a quantum measurement that includes a detecting device, it is possible to construct from the…
We propose a randomized polynomial time algorithm for computing nontrivial zeros of quadratic forms in 4 or more variables over $\mathbb{F}_q(t)$, where $\mathbb{F}_q$ is a finite field of odd characteristic. The algorithm is based on a…
The paper is devoted to quantization of polynomial momentum observables in the cotangent bundle of a smooth manifold. A quantization procedure is proposed allowing to quantize a wide class of functions which are polynomials of any order in…
Several recently discovered properties of multiple families of special polynomials (some orthogonal and some not) that satisfy certain differential, difference or q-difference equations are reviewed. A general method of construction of…
We study several related problems on polynomials with integer coefficients. This includes the integer Chebyshev problem, and the Schur problems on means of algebraic numbers. We also discuss interesting applications to approximation by…
Let K be F_q((T)), or more generally any field of characteristic p equipped with a valuation having a finite residue field of q elements. Then a polynomial f(x) in K[x] having k+1 nonzero coefficients has at most q^k distinct zeros in K. We…
Quantum algorithms can be analyzed in a query model to compute Boolean functions where input is given in a black box, but the aim is to compute function value for arbitrary input using as few queries as possible. In this paper we…
Quantum robots are described as mobile quantum computers and ancillary systems that move in and interact with arbitrary environments. Their dynamics is given as tasks which consist of sequences of alternating computation and action phases.…
We propose a quantum algorithm for solving the following problem: given the Hamiltonian of a physical system and one of its eigenvalues, how to obtain the corresponding eigenstate? The algorithm is based on the resonance phenomena. For a…
Quantum search is a quantum mechanical technique for searching N possibilities in only sqrt(N) steps. This has been proved to be the best possible algorithm for the exhuastive search problem in the sense the number of queries it requires…
We prove that the results of a finite set of general quantum measurements on an arbitrary dimensional quantum system can be simulated using a polynomial (in measurements) number of hidden-variable states. In the limit of infinitely many…
Quantum error correction allows for faulty quantum systems to behave in an effectively error free manner. One important class of techniques for quantum error correction is the class of quantum subsystem codes, which are relevant both to…