Uncertainty Analysis and Order-by-Order Optimization of Chiral Nuclear Interactions

Chiral effective field theory (chi EFT) provides a systematic approach to describe low-energy nuclear forces. Moreover, chi EFT is able to provide well-founded estimates of statistical and systematic uncertainties-although this unique advantage has not yet been fully exploited. We fill this gap by performing an optimization and statistical analysis of all the low-energy constants (LECs) up to next-to-next-to-leading order. Our optimization protocol corresponds to a simultaneous fit to scattering and bound-state observables in the pion-nucleon, nucleon-nucleon, and few-nucleon sectors, thereby utilizing the full model capabilities of chi EFT. Finally, we study the effect on other observables by demonstrating forward-error-propagation methods that can easily be adopted by future works. We employ mathematical optimization and implement automatic differentiation to attain efficient and machine-precise first-and second-order derivatives of the objective function with respect to the LECs. This is also vital for the regression analysis. We use power-counting arguments to estimate the systematic uncertainty that is inherent to chi EFT, and we construct chiral interactions at different orders with quantified uncertainties. Statistical error propagation is compared with Monte Carlo sampling, showing that statistical errors are, in general, small compared to systematic ones. In conclusion, we find that a simultaneous fit to different sets of data is critical to (i) identify the optimal set of LECs, (ii) capture all relevant correlations, (iii) reduce the statistical uncertainty, and (iv) attain order-by-order convergence in chi EFT. Furthermore, certain systematic uncertainties in the few-nucleon sector are shown to get substantially magnified in the many-body sector, in particular when varying the cutoff in the chiral potentials. The methodology and results presented in this paper open a new frontier for uncertainty quantification in ab initio nuclear theory

Rake, Peel, Sketch:The Signal Processing Pipeline Revisited

The prototypical signal processing pipeline can be divided into four blocks. Representation of the signal in a basis suitable for processing. Enhancement of the meaningful part of the signal and noise reduction. Estimation of important statistical properties of the signal. Adaptive processing to track and adapt to changes in the signal statistics. This thesis revisits each of these blocks and proposes new algorithms, borrowing ideas from information theory, theoretical computer science, or communications. First, we revisit the Walsh-Hadamard transform (WHT) for the case of a signal sparse in the transformed domain, namely that has only K †N non-zero coefficients. We show that an efficient algorithm exists that can compute these coefficients in O(K log2(K) log2(N/K)) and using only O(K log2(N/K)) samples. This algorithm relies on a fast hashing procedure that computes small linear combinations of transformed domain coefficients. A bipartite graph is formed with linear combinations on one side, and non-zero coefficients on the other. A peeling decoder is then used to recover the non-zero coefficients one by one. A detailed analysis of the algorithm based on error correcting codes over the binary erasure channel is given. The second chapter is about beamforming. Inspired by the rake receiver from wireless communications, we recognize that echoes in a room are an important source of extra signal diversity. We extend several classic beamforming algorithms to take advantage of echoes and also propose new optimal formulations. We explore formulations both in time and frequency domains. We show theoretically and in numerical simulations that the signal-to-interference-and-noise ratio increases proportionally to the number of echoes used. Finally, beyond objective measures, we show that echoes also directly improve speech intelligibility as measured by the perceptual evaluation of speech quality (PESQ) metric. Next, we attack the problem of direction of arrival of acoustic sources, to which we apply a robust finite rate of innovation reconstruction framework. FRIDA â the resulting algorithm â exploits wideband information coherently, works at very low signal-to-noise ratio, and can resolve very close sources. The algorithm can use either raw microphone signals or their cross- correlations. While the former lets us work with correlated sources, the latter creates a quadratic number of measurements that allows to locate many sources with few microphones. Thorough experiments on simulated and recorded data shows that FRIDA compares favorably with the state-of-the-art. We continue by revisiting the classic recursive least squares (RLS) adaptive filter with ideas borrowed from recent results on sketching least squares problems. The exact update of RLS is replaced by a few steps of conjugate gradient descent. We propose then two different precondi- tioners, obtained by sketching the data, to accelerate the convergence of the gradient descent. Experiments on artificial as well as natural signals show that the proposed algorithm has a performance very close to that of RLS at a lower computational burden. The fifth and final chapter is dedicated to the software and hardware tools developed for this thesis. We describe the pyroomacoustics Python package that contains routines for the evaluation of audio processing algorithms and reference implementations of popular algorithms. We then give an overview of the microphone arrays developed

Making predictions using χEFT

In this thesis we explore the merits of chiral effective field theory (χEFT) as a model for low-energy nuclear physics. χEFT is an effective field theory based on quantum chromo dynamics (QCD) describing low-energy interactions of nucleons and pions. We estimate the inherent uncertainties of χEFT and the accompanying methods used to compute observables in order to test the predictive power of the model. We use experimental pion-nucleon, nucleon- nucleon and few-nucleon data to perform a simultaneous fit of the low-energy constants in χEFT. This results in small statistical uncertainties in the model. The results show a clear order-by-order improvement of χEFT with the systematical model error dominating the total error budget

Quantum walk and Wigner function on a lattice

La informació quàntica és un camp relativament jove de la Física, que té com a objectiu explorar les lleis de la mecànica quàntica per a la transmissió i el processament de la informació. Com a exemple d’aplicacions es poden esmentar les comunicacions segures, basades en la distribució de clau quàntica, i algoritmes quàntics que superen als seus homàlegs clàssics per a un determinat nombre de problemes. A més, les eines desenvolupades en el context de la informació quàntica han demostrat ser de gran utilitat per aprofundir en la comprensió dels sistemes quàntics, per exemple, en el context dels problemes de molts cossos quàntics. Una de les principals aplicacions de la potència de la mecànica quàntica en tasques computacionals ́és la manipulació de sistemes quàntics al laboratori per tal de realitzar simulacions quàntiques, i els diferents estudis experimentals s’estan realitzant en l’actualitat cap aquest objectiu. Especialment prometedores són les primeres simulacions quàntiques de sistemes atòmics ultrafreds atrapats en xarxes optiques, on els resultats superen els càlculs clàssics. Aquesta tesi aplica eines d’informació quàntica a la descripció i l’estudi de diversos sistemes quàntics i a processos que succeeixen en un espai discret, és a dir, en una xarxa. Fins i tot una sola partícula quàntica amb spin 1/2 pot donar lloc a fenomens que difereixen de forma radical de qualsevol analogia clàssica. En alguns casos, la nostra comprensió dels processos físics és més intuïtiva per al cas continu, i per tant, el nostre estudi es connecta fins al límit continu adequat. La tesi s’estructura en dues parts. La primera d’elles s’emmarca en l’estudi i comprensió d’un algoritme quàntic en particular, el passeig quàntic. Per tal d’explotar el passeig quàntic i aplicar-lo a la construcció d’algoritmes quàntics, és important entendre i controlar el seu comportament tant com siga possible. Una de les característiques analitzades en aquesta tesi és el passeig quàntic discret en N dimensions des de la perspectiva de les relacions de dispersió. Fent ús de condicions inicials esteses en l’espai de posicions, s’obté una equació d’ona en el límit continu. Aquesta equació ens permet d’entendre algunes propietats conegudes i dissenyar interessants comportaments. Apliquem l’estudi al passeig quàntic en dos i tres dimensions per a la moneda de Grover, on la relació dedispersió presenta punts i interseccions particulars on la dinàmica és especialment diferent. D’altra banda, s’analitza el comportament del passeig quàntic com un procés Markovià. Amb aquest objectiu, s’investiga l’evolució temporal de la matiu densitat reduïda per un passeig quàntic de temps discret en una xarxa unidimensional. S’analitza la dinàmica de la matriu densitat reduïda en el cas estàndard, sense decoherència, i quan el sistema està exposat als efectes de decoherència. Analitzem el comportament Markovià de l’evolució en el sentit definit en [1] examinant la distància de traça per a possibles parells de estats inicials com una funció del temps. Arribem a la conclusió que l’evolució de la matriu densitat reduïda en el cas lliure és no Markoviana i, quan el nivell de soroll augmenta, la dinàmica s’aproxima a un procés Markovià. La segona part d’aquesta tesi proposa una generalització de la coneguda funció de Wigner per a una partícula que es mou en una xarxa infinita en una dimensió. L’estudi de la mecànica quàntica en l’espai de fases a través de les distribucions de quasi-probabilitat s’aplica en molts camps de la física i la funció de Wigner és probablement la més utilitzada. S’estudia la funció de Wigner per a un sistema quàntic en un espai d’Hilbert discret, de dimensió infinita, tal com una partícula sense spin en moviment en una xarxa infinita unidimensional. Es discuteixen les peculiaritats d’aquest escenari i la construcció de l’espai fàsic associat, i es proposa una definició significativa de la funció de Wigner en aquest cas, a més es caracteritza el conjunt d’estats purs per als quals la funció de Wigner és no negativa. També ampliem la definició proposada per incloure un grau intern de llibertat, com ara l’spin. La dinàmica d’una partícula en una xarxa amb, i sense spin, en diferents casos, també s’analitza en termes de la funció de Wigner corresponent. Mostrem solucions explícites en el cas d’evolució hamiltoniana sota un potencial depenent de la posició que pot incloure un acoblament d’spin, i per a l’evolució governada per una equació mestra sota alguns simples models de decoherència. Proposem una mesura de la no-classicitat dels estats en un sistema amb un espai d’Hilbert discret i infinit que és consistent amb el límit continu. I, en darrer lloc, discutim la possibilitat d’ampliar el concepte de negativitat de la funció de Wigner al cas en el qual s’inclou el grau de llibertad d’spin.Quantum information is a relatively young field of Physics, that aims to exploit the laws of quantum mechanics for the transmission and processing of information. As illustrative applications one can mention secure communications, based on quantum key distribution, and quantum algorithms that outperform their classical counterparts for a number of problems. Furthermore, the tools developed in the context of Quantum Information have proven extremely useful to deepen the understanding of quantum systems, for instance in the context of quantum many-body problems. One of the main applications of the power of quantum mechanics to computational tasks is the manipulation of quantum systems in the lab in order to perform quantum simulations, and different experimental approaches are currently being pursued towards this goal. Especially promising are ultracold atomic systems trapped in optical lattices, where the first quantum simulations that outperform the feasible classical calculations have already been realized. This thesis applies quantum information tools to the description and the study of several quantum systems and processes that happen on a discrete space, i.e. on a lattice. Even a single quantum particle with spin 1/2 hopping on a lattice can give rise to phenomena that dramatically differ from any classical analogy. In some cases, our understanding of the physical processes is more intuitive for the continuous case, and hence we connect our study to the proper continuum limit. The thesis is structured in two parts. The first one is framed within the study and understanding of a particular quantum algorithm, namely the quantum walk. In order to exploit the quantum walk and apply it to the construction of quantum algorithms, it is important to understand and control its behavior as much as possible. One of the features analyzed in this thesis is the discrete time quantum walk in N dimensions from the perspective of its dispersion relations. Making use of the spatially extended initial conditions, a wave equation in the continuum limit is obtained. This equation allows us to understand some known properties, and to design interesting behaviors. We apply the study to the two and three dimensional Grover quantum walk, where the dispersion relation presents particular points and intersections where the dynamics is specially distinct. On the other hand, we analyze the behavior of the quantum walk as a Markovian process. With this aim, we investigate the time evolution of the chirality reduced density matrix for a discrete time quantum walk on a one-dimensional lattice. We analyze the dynamics of the reduced density matrix in the standard case, without decoherence, and when the system is exposed to the effects of decoherence. We analyze the Markovian behavior in the sense defined in [1] examining the trace distance for possible pairs of initial states as a function of time which gives us the distinguishability of two states and it is related with the Markovian behavior of the system. We conclude that the evolution of the reduced density matrix in the free case is non-Markovian and, as the level of noise increases, the dynamics approaches a Markovian process. The second part of this thesis proposes a generalization of the known Wigner function for a particle moving on an infinite lattice in one dimension. The study of the quantum mechanics in phase space through quasi-probability distributions is applied in many fields of physics and the Wigner function is probably the most commonly used one. We study the Wigner function for a quantum system with a discrete, infinite dimensional Hilbert space, such as a spinless particle moving on a one dimensional infinite lattice. We discuss the peculiarities of this scenario and of the associated phase space construction, propose a meaningful definition of the Wigner function in this case, and characterize the set of pure states for which it is non-negative. We also extended the proposed definition to include an internal degree of freedom, such as the spin. The dynamics of a particle on a lattice with and without spin in different cases are also analyzed in terms of the corresponding Wigner function. We show explicit solutions for the case of Hamiltonian evolution under a position dependent potential that may include a spin coupling, and for the evolution governed by a master equation under some simple models of decoherence. We propose a measure of non-classicality for states in the system with a discrete infinite dimensional Hilbert space which is consistent with the continuum limit. And we discuss the possibility of extending a negativity concept for the Wigner function in the case in which the spin degree of freedom is included