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José M. Tornero

Publications and source records attributed to José M. Tornero.

14 recordsLinked to original sources
arXiv2026

Two Problems on Quantum Computing in Finite Abelian Groups

In the context of finite Abelian groups two problems are presented and solved using quantum computing techniques. The first is the well--known Hidden Subgroup Problem, originally solved by Simon in a landmark work. The second is the Fully Balanced Image Problem, originally introduced by the authors (joint with J. Ossorio--Castillo), which is related to a certain class of mappings (which contains strictly, for instance, the family of group morphisms). Both problems are tackled using a combination of two techniques: first, a conversion into Boolean objects, better suited for quantum computing arguments, and subsequently a custom--tailored algorithm which takes advantage of the Generalised Phase--Kick Back technique.

quant-ph↗
arXiv2026

On the torsion growth in quadratic number fields for elliptic curves defined over the rationals

Given an elliptic curve defined over the field of rational numbers, it is known how its torsion subgroup may grow when we make a base change to a quadratic number field. In this paper we consider the inverse question: if we have the elliptic curve defined over the rationals and we know how the torsion subgroup grows, what can we say about the field? Our main result gives an explicit relationship between the primes dividing the conductor of the curve and the conductor of the extension as a first approach to a better understanding of this problem.

math.NT↗
arXiv2025

Quantum computing from a mathematical perspective: a description of the quantum circuit model

This paper is an essentially self-contained and rigorous description of the fundamental principles of quantum computing from a mathematical perspective. It is intended to help mathematicians who want to get a grasp of this quickly growing discipline and find themselves taken aback by the language gap between mathematics and the pioneering fields on the matter: computer science and quantum physics. A first version of this paper was originally published at arXiv in October 2018. Minor corrections added on Spring 2025.

quant-ph↗
arXiv2025

Onthe computation of the MED closure of a numerical semigroup

Maximally embedding dimension (MED) numerical semigroups are a wide and interesting family, with some remarkable algebraic and combinatorial properties. Associated to any numerical semigroup one can construct a MED closure, as it is well known. This paper shows two different explicit methods to construct this closure which also sheds new light on the very nature of this object.

math.CO↗
arXiv2024

Further Applications of the Generalised Phase Kick-Back

In our previous work, we defined a quantum algorithmic technique known as the Generalised Phase Kick-Back, or $GPK$, and analysed its applications in generalising some classical quantum problems, such as the Deutsch-Jozsa problem or the Bernstein-Vazirani problem. We also proved that using this technique we can solve Simon's problem in a more efficient manner. In this paper we continue analysing the potential of this technique, defining the concept of $\mathbf{y}$-balanced functions and solving a new problem, which further generalises the generalised Deutsch-Jozsa problem (the fully balanced image problem). This problem also underlines the relation between quantum computation and Boolean function theory, and, in particular, the Walsh and Fourier-Hadamard transforms. We finish our discussion by solving the generalised version of Simon's problem using the $GPK$ algorithm, while analysing the efficiency of this new solution.

quant-ph↗
arXiv2024

Blurred combinatorics in resolution of singularities: (a little) beyond the characteristic polytope

We introduce a variation of the well-known Newton-Hironaka polytope for algebroid hypersurfaces. This combinatorial object is a perturbed version of the original one, parametrized by a real number. For well-chosen values of the parameter, the objects obtained are very close to the original, while at the same time presenting more (hopefully interesting) information in a way which does not depend on the choice of parameter.

math.AG↗
arXiv2022

A Generalisation of the Phase Kick-Back

In this paper, we present a generalisation of the Phase Kick-Back technique, which is central to some of the classical algorithms in quantum computing, such as the Deutsch--Jozsa algorithm, Simon's algorithm or Grover's algorithm. We will begin by recalling the Phase Kick-Back technique to then introduce the new generalised version and analyse it. After that, we will present a new generalised version of the Deutsch--Jozsa problem and it will be solved using the previously defined technique. Finally, we will present a generalised version of the Bernstein-Vazirani problem and solve it using this technique to better understand its inner workings.

quant-ph↗
arXiv2019

An adiabatic quantum algorithm for the Frobenius problem

The (Diophantine) Frobenius problem is a well-known NP-hard problem (also called the stamp problem or the chicken nugget problem) whose origins lie in the realm of combinatorial number theory. In this paper we present an adiabatic quantum algorithm which solves it, using the so-called Apéry set of a numerical semigroup, via a translation into a QUBO problem. The algorithm has been specifically designed to run in a D-Wave 2X machine.

math.CO↗
arXiv2019

Characterization of Gaps and Elements of a Numerical Semigroup Using Groebner Bases

This article is partly a survey and partly a research paper. It tackles the use of Groebner bases for addressing problems of numerical semigroups, which is a topic that has been around for some years, but it does it in a systematic way which enables us to prove some results and a hopefully interesting characterization of the elements of a semigroup in terms of Groebner bases.

math.CO↗
arXiv2019

On the computation of the Apéry set of numerical monoids and affine semigroups

A simple way of computing the Apéry set of a numerical semigroup (or monoid) with respect to a generator, using Groebner bases, is presented, together with a generalization for affine semigroups. This computation allows us to calculate the type set and, henceforth, to check the Gorenstein condition which characterizes the symmetric numerical subgroups.

math.CO↗
arXiv2019

Integral points in rational polygons: a numerical semigroup approach

In this paper we use an elementary approach by using numerical semigroups (specifically, those with two generators) to give a formula for the number of integral points inside a right-angled triangle with rational vertices. This is the basic case for computing the number of integral points inside a rational (not necessarily convex) polygon.

math.CO↗
arXiv2019

Combinatorics and their evolution in resolution of embedded algebroid surfaces

The seminal concept of characteristic polygon of an embedded algebroid surface, first developed by Hironaka, seems well suited for combinatorially (perhaps even effectively) tracking of a resolution process. However, the way this object evolves through the resolution of singularities was not really well understood, as some references had pointed out. The aim of this paper is to explain, in a clear way, how this object changes as the surface gets resolved. In order to get a precise description of the phenomena involved, we need to use different techniques and ideas. Eventually, some effective results regarding the number of blow-ups needed to decrease the multiplicity are obtained as a side product.

math.AG↗
arXiv2013

Markoff-Rosenberger triples in arithmetic progression

We study the solutions of the Rosenberg--Markoff equation ax^2+by^2+cz^2 = dxyz (a generalization of the well--known Markoff equation). We specifically focus on looking for solutions in arithmetic progression that lie in the ring of integers of a number field. With the help of previous work by Alvanos and Poulakis, we give a complete decision algorithm, which allows us to prove finiteness results concerning these particular solutions. Finally, some extensive computations are presented regarding two particular cases: the generalized Markoff equation x^2+y^2+z^2 = dxyz over quadratic fields and the classic Markoff equation x^2+y^2+z^2 = 3xyz over an arbitrary number field.

math.NT↗