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Mario Ponce

Publications and source records attributed to Mario Ponce.

At least 19 recordsLinked to original sources
arXiv2024

Performance and scaling analysis of variational quantum simulation

We present an empirical analysis of the scaling of the minimal quantum circuit depth required for a variational quantum simulation (VQS) method to obtain a solution to the time evolution of a quantum system within a predefined error tolerance. In a comparison against a non-variational method based on Trotterized time evolution, we observe a better scaling of the depth requirements using the VQS approach with respect to both the size of the system and the simulated time. Results are also put into perspective by discussing the corresponding classical complexity required for VQS. Our results allow us to identify a possible advantage region for VQS over Trotterization.

quant-ph↗
arXiv2024

Odometers in non-compact spaces

We define an odometer in the Baire space. That is the non-compact space of one sided sequences of natural numbers. We go on to prove that it is topologically conjugated to the dyadic odometer restricted to an appropriate non-compact subset of the space of one sided sequences on two symbols. We extend the action of the odometer to finite words. It turns out that, arranging the finite words in appropriate binary tree, this extension runs through the tree from left-right and top-down. Several well known binary trees made out of rational numbers, such as the Kepler tree or the Calkin and Wilf tree, are recovered in this way. Indeed, it suffices to identify finite words with rational numbers by means of certain continued fractions. The odometric action allow us to recover, in a unified way, several counting results for the rational numbers. Moreover, associated to certain interval maps with countably many branches we construct their corresponding odometers. Explicit formulas are provided in the cases of the Gauss map (continued fractions) and the Renyi map (backward continued fractions).

math.DS↗
arXiv2023

Odometers, Backward continued fractions and counting rationals

It has been more than twenty years since Moshe Newman, based on work by Neil Calkin and Herbert Wilf, introduced an explicit bijection between the rational and natural numbers. Interestingly, this bijection is dynamic in nature. Indeed, Newman's map has the property that the orbit of zero provides the required bijection. Claudio Bonanno and Stefano Isola, using continued fractions expansions, described the dynamics of its first return time map T. They proved that it is topologically conjugated to the dyadic odometer. In this article, we prove that the correct numerical system needed to analyze this map is the backward continued fractions. Indeed, this approach has the advantage that it provides explicitly the action of T on the expansion. As a by-product, we naturally obtain an explicit formula for Minkowski's question mark function in terms of backward continued fractions. The whole point of Newman was to provide an explicit bijection, our approach shares the same taste for the explicit.

math.DS↗
arXiv2023

Benchmarking Digital-Analog Quantum Computation

Digital-Analog Quantum Computation (DAQC) has recently been proposed as an alternative to the standard paradigm of digital quantum computation. DAQC creates entanglement through a continuous or analog evolution of the whole device, rather than by applying two-qubit gates. This manuscript describes an in-depth analysis of DAQC by extending its implementation to arbitrary connectivities and by performing the first systematic study of its scaling properties. We specify the analysis for three examples of quantum algorithms, showing that except for a few specific cases, DAQC is in fact disadvantageous with respect to the digital case.

quant-ph↗
arXiv2023

Optimal, hardware native decomposition of parameterized multi-qubit Pauli gates

We show how to efficiently decompose a parameterized multi-qubit Pauli (PMQP) gate into native parameterized two-qubit Pauli (P2QP) gates minimizing both the circuit depth and the number of P2QP gates. Given a realistic quantum computational model, we argue that the technique is optimal in terms of the number of hardware native gates and the overall depth of the decomposition. Starting from PMQP gate decompositions for the path and star hardware graph, we generalize the procedure to any generic hardware graph and provide exact expressions for the depth and number of P2QP gates of the decomposition. Furthermore, we show how to efficiently combine the decomposition of multiple PMQP gates to further reduce the depth as well as the number of P2QP gates for a combinatorial optimization problem using the Lechner-Hauke-Zoller (LHZ) mapping.

quant-ph↗
arXiv2021

The Hal\'asz-Sz\'ekely Barycenter

We introduce a notion of barycenter of a probability measure related to the symmetric mean of a collection of nonnegative real numbers. Our definition is inspired by the work of Hal\'asz and Sz\'ekely, who in 1976 proved a law of large numbers for symmetric means. We study analytic properties of this Hal\'asz-Sz\'ekely barycenter. We establish fundamental inequalities that relate the symmetric mean of a list of nonnegative real numbers with the barycenter of the measure uniformly supported on these points. As consequence, we go on to establish an ergodic theorem stating that the symmetric means of a sequence of dynamical observations converges to the Hal\'asz-Sz\'ekely barycenter of the corresponding distribution.

math.PR↗
arXiv2020

On the projective derivative cocycle for circle diffeomorphisms

We study the projective derivative as a cocycle of M\"obius transformations over groups of circle diffeomorphisms. By computing precise expressions for this cocycle, we obtain several results about reducibility and almost reducibility to a cocycle of rotations. We also introduce an extension of this cocycle to the diagonal action on the 3-torus for which we generalize the previous results.

math.DS↗
arXiv2016

Perfect Matchings in Inhomogeneous Random Bipartite Graphs in Random Environment

In this note we study inhomogeneous random bipartite graphs in random environment. These graphs can be thought of as an extension of the classical Erd\"os-R\'enyi random graphs in a random environment. We show that the expected number of perfect matchings obeys a precise quenched asymptotic and that it can be approximated using an iterative process that converges exponentially fast.

math.CO↗
arXiv2015

The Scaling Mean and a Law of Large Permanents

In this paper we study two types of means of the entries of a nonnegative matrix: the \emph{permanental mean}, which is defined using permanents, and the \emph{scaling mean}, which is defined in terms of an optimization problem. We explore relations between these two means, making use of important results by Ergorychev and Falikman (the van~der~Waerden conjecture), Friedland, Sinkhorn, and others. We also define a scaling mean for functions in a much more general context. Our main result is a Law of Large Permanents, a pointwise ergodic theorem for permanental means of dynamically defined matrices that expresses the limit as a functional scaling mean. The concepts introduced in this paper are general enough so to include as particular cases certain classical types of means, as for example symmetric means and Muirhead means. As a corollary, we reobtain a formula of Hal\'asz and Sz\'ekely for the limit of the symmetric means of a stationary random process.

math.DS↗
arXiv2014

An ergodic theorem for permanents of oblong matrices

We form a sequence of oblong matrices by evaluating an integrable vector-valued function along the orbit of an ergodic dynamical system. We obtain an almost sure asymptotic result for the permanents of those matrices. We also give an application to symmetric means.

math.DS↗
arXiv2014

Regularity of Mediatrices in Surfaces

For distinct points $p$ and $q$ in a two-dimensional Riemannian manifold, one defines their mediatrix $L_{pq}$ as the set of equidistant points to $p$ and $q$. It is known that mediatrices have a cell decomposition consisting of a finite number of branch points connected by Lipschitz curves. This paper establishes additional geometric regularity properties of mediatrices. We show that mediatrices have the radial linearizability property, which implies that at each point they have a geometrically defined derivative in the branching directions. Also, we study the particular case of mediatrices on spheres, by showing that they are Lipschitz simple closed curves exhibiting at most countably many singularities, with finite total angular deficiency.

math.DG↗
arXiv2012

Hyperbolization of cocycles by isometries of the euclidean space

We study hyperbolized versions of cohomological equations that appear with cocycles by isometries of the euclidean space. These (hyperbolized versions of) equations have a unique continuous solution. We concentrate in to know whether or not these solutions converge to a genuine solution to the original equation, and in what sense we can use them as good approximative solutions. The main advantage of considering solutions to hyperbolized cohomological equations is that they can be easily described, since they are global attractors of a naturally defined skew-product dynamics. We also include some technical results about twisted Birkhoff sums and exponential averaging.

math.DS↗
arXiv2012

Old and new about equidistant sets and generalized conics

This article is devoted to the study of classical and new results concerning equidistant sets, both from the topological and metric point of view. We start with a review of the most interesting known facts about these sets in the euclidean space and then we prove that equidistant sets vary continuously with their focal sets. In the second part we propose a viewpoint in which equidistant sets can be thought of as natural generalization for conics. Along these lines, we show that many geometric features of classical conics can be retrieved in more general equidistant sets. In the Appendix we prove a shadowing property of equidistant sets and provide sharp estimates. This result should be of interest for computer simulations.

math.MG↗
arXiv2011

A Livsic type theorem for germs of analytic diffeomorphisms

We deal with the problem of the validity of Livsic's theorem for cocycles of diffeomorphisms satisfying the orbit periodic obstruction over an hyperbolic dynamics. We give a result in the positive direction for cocycles of germs of analytic diffeomorphisms at the origin.

math.DS↗
arXiv2010

Towards a semilocal study of parabolic invariant curves for fibred holomorphic maps

We introduce the study of the local dynamics around a parabolic indifferent invariant curve for fibred holomorphic maps. As in the classical non-fibred case, we show that petals are the main ingredient. Nevertheless, one expects the properties of the base rotation number should play an important role in the arrangement of the petals. We exhibit examples where the existence and the number of petals depend not just on the complex coordinate of the map, but on the base rotation number. Furthermore, under additional hypothesis on the arithmetics and smoothness of the map, we present a theorem that allows to characterize the local dynamics around a parabolic invariant curve.

math.DS↗