Harmonic (Quantum) Neural Networks

Harmonic functions are abundant in nature, appearing in limiting cases of Maxwell's, Navier-Stokes equations, the heat and the wave equation. Consequently, there are many applications of harmonic functions from industrial process optimisation to robotic path planning and the calculation of first exit times of random walks. Despite their ubiquity and relevance, there have been few attempts to incorporate inductive biases towards harmonic functions in machine learning contexts. In this work, we demonstrate effective means of representing harmonic functions in neural networks and extend such results also to quantum neural networks to demonstrate the generality of our approach. We benchmark our approaches against (quantum) physics-informed neural networks, where we show favourable performance.Comment: 12 pages (main), 7 pages (supplementary), 7 figure

Qadence: a differentiable interface for digital-analog programs

Digital-analog quantum computing (DAQC) is an alternative paradigm for universal quantum computation combining digital single-qubit gates with global analog operations acting on a register of interacting qubits. Currently, no available open-source software is tailored to express, differentiate, and execute programs within the DAQC paradigm. In this work, we address this shortfall by presenting Qadence, a high-level programming interface for building complex digital-analog quantum programs developed at Pasqal. Thanks to its flexible interface, native differentiability, and focus on real-device execution, Qadence aims at advancing research on variational quantum algorithms built for native DAQC platforms such as Rydberg atom arrays

Simulations de Monte Carlo quantique améliorées : de systèmes ouverts aux solides cristallins

In this thesis we present algorithmic progresses as well as applications of continuum quantum Monte Carlo (QMC) methods for electronic structure calculations by first principles. The improvements we propose allow to tackle much larger molecular as well as extended systems by QMC, with the ultimate goal of making QMC a valid alternative to density functional theory (DFT). All results have been obtained with the TurboRVB software, which we contributed to develop. At first, we present a QMC framework based on the Jastrow-Geminal wavefunction which combines great flexibility with a compact analytical form, while providing at the same time an accurate treatment of electron correlations. We apply an original atomic embedding scheme for reducing the basis set size to the water molecule and to a simple model of proton transfer (PT) in aqueous systems. Our results pave the way to the study of microscopic phenomena such as PT directly by QMC. Afterwards, we extend our QMC framework in order to simulate crystalline solids. We propose a novel procedure to find special values of the boundary conditions which allow to greatly reduce the finite-size errors affecting solid state QMC simulations. Using the techniques previously developed, we study the iron-based superconductor FeSe. We show that QMC provides the best crystal structure predictions on this compound; by means of a systematic study of the energy landscape at different magnetic orderings, we show a strong link between structural, magnetic and charge degrees of freedom in FeSe. Our results represent an important step towards a quantitative understanding of high-temperature superconductivity by first-principles.Dans cette thèse nous présentons des progrès algorithmiques ainsi que plusieurs applications des méthodes de Monte Carlo quantique (QMC) pour simulations à partir des premiers principes. Les améliorations que nous proposons permettent d'étudier par QMC des systèmes de plus grosse taille voire périodiques, avec l'ambition de faire du QMC une alternative valable à la théorie de la fonctionnelle de la densité (DFT). Tous les résultats ont été obtenus par le logiciel TurboRVB. D'abord, nous présentons une implémentation du QMC basée sur la fonction d'onde Jastrow-Geminale qui combine une grande flexibilité avec un traitement précis des corrélations électroniques. On a appliqué une technique originale de plongement pour réduire la taille de la base atomique à la molécule d'eau ainsi qu'à un modèle simplifié du transfert de protons (TP) dans l'eau. Nos résultats ouvrent la voie à l'étude des phénomènes microscopiques tels que le TP directement par QMC. Ensuite, on a amélioré notre méthode afin de simuler les solides cristallins. Grâce à une nouvelle procédure pour choisir de manière appropriée les conditions aux limites, nous avons pu réduire les erreurs de taille finie qui affectent les simulations QMC des solides. Sur la base des techniques développées, nous étudions enfin le supraconducteur FeSe. Le QMC fournit le meilleur résultat concernant sa structure cristalline; via une étude systématique du paysage énergétique à différentes configurations magnétiques, nous montrons un lien fort entre la structure, le magnétisme et les mouvements de charge dans ce matériau, prélude à une compréhension quantitative de la supraconductivité à haute température des premiers principes

Competing collinear magnetic structures in superconducting FeSe by first-principles quantum Monte Carlo calculations

Resolving the interplay between magnetic interactions and structural properties in strongly correlated materials through a quantitatively accurate approach has been a major challenge in condensed-matter physics. Here we apply highly accurate first-principles quantum Monte Carlo (QMC) techniques to obtain structural and magnetic properties of the iron selenide (FeSe) superconductor under pressure. Where comparable, the computed properties are very close to the experimental values. Of potential ordered magnetic configurations, collinear spin configurations are the most energetically favorable over the explored pressure range. They become nearly degenerate in energy with bicollinear spin orderings at around 7 GPa, when the experimental critical temperature Tc is the highest. On the other hand, ferromagnetic, checkerboard, and staggered dimer configurations become relatively higher in energy as the pressure increases. The behavior under pressure is explained by an analysis of the local charge compressibility and the orbital occupation as described by the QMC many-body wave function, which reveals how spin, charge, and orbital degrees of freedom are strongly coupled in this compound. This remarkable pressure evolution suggests that stripelike magnetic fluctuations may be responsible for the enhanced Tc in FeSe and that higher Tc is associated with nearness to a crossover between collinear and bicollinear ordering

Exact special twist method for quantum Monte Carlo simulations

We present a systematic investigation of the special twist method introduced by Rajagopal et al. [Phys. Rev. B 51, 10591 (1995)] for reducing finite-size effects in correlated calculations of periodic extended systems with Coulomb interactions and Fermi statistics. We propose a procedure for finding special twist values which, at variance with previous applications of this method, reproduce the energy of the mean-field infinite-size limit solution within an adjustable (arbitrarily small) numerical error. This choice of the special twist is shown to be the most accurate single-twist solution for curing one-body finite-size effects in correlated calculations. For these reasons we dubbed our procedure \u201cexact special twist\u201d (EST). EST only needs a fully converged independent-particles or mean-field calculation within the primitive cell and a simple fit to find the special twist along a specific direction in the Brillouin zone. We first assess the performances of EST in a simple correlated model such as the three-dimensional electron gas. Afterwards, we test its efficiency within ab initio quantum Monte Carlo simulations of metallic elements of increasing complexity. We show that EST displays an overall good performance in reducing finite-size errors comparable to the widely used twist average technique but at a much lower computational cost since it involves the evaluation of just one wave function. We also demonstrate that the EST method shows similar performances in the calculation of correlation functions, such as the ionic forces for structural relaxation and the pair radial distribution function in liquid hydrogen. Our conclusions point to the usefulness of EST for correlated supercell calculations; our method will be particularly relevant when the physical problem under consideration requires large periodic cells

TurboRVB: a many-body toolkit for ab initio electronic simulations by quantum Monte Carlo

International audienceTurboRVB is a computational package for ab initio Quantum Monte Carlo (QMC) simulations of both molecular and bulk electronic systems. The code implements two types of well established QMC algorithms: Variational Monte Carlo (VMC), and Diffusion Monte Carlo in its robust and efficient lattice regularized variant. A key feature of the code is the possibility of using strongly correlated many-body wave functions, capable of describing several materials with very high accuracy, even when standard mean-field approaches (e.g., density functional theory (DFT)) fail. The electronic wave function (WF) is obtained by applying a Jastrow factor, which takes into account dynamical correlations, to the most general mean-field ground state, written either as an antisymmetrized geminal product with spin-singlet pairing, or as a Pfaffian, including both singlet and triplet correlations. This wave function can be viewed as an efficient implementation of the so-called resonating valence bond (RVB) ansatz, first proposed by L. Pauling and P. W. Anderson in quantum chemistry and condensed matter physics, respectively. The RVB ansatz implemented in TurboRVB has a large variational freedom, including the Jastrow correlated Slater determinant as its simplest, but nontrivial case. Moreover , it has the remarkable advantage of remaining with an affordable computational cost, proportional to the one spent for the evaluation of a single Slater determinant. Therefore, its application to large systems is computationally feasible. The WF is expanded in a localized basis set. Several basis set functions are implemented, such as Gaussian, Slater, and mixed types, with no restriction on the choice of their contraction. The code implements the adjoint algorithmic differentiation that enables a very efficient evaluation of energy derivatives, comprising the ionic forces. Thus, one can perform structural optimizations and molecular dynamics in the canonical NVT ensemble at the VMC level. For the electronic part, a full WF optimization (Jastrow and antisymmetric parts together) is made possible thanks to state-of-the-art stochastic algorithms for energy minimization. In the optimization procedure, the first guess can be obtained at the mean-field level by a built-in DFT driver. The code has been efficiently parallelized by using a hybrid MPI-OpenMP protocol, that is also an ideal environment for exploiting the computational power of modern GPU accelerators

Quantum Monte Carlo Study of the Protonated Water Dimer

We report an extensive theoretical study of the protonated water dimer H5O2+ (Zundel ion) by means of the highly correlated variational Monte Carlo and lattice regularized Monte Carlo approaches. This system represents the simplest model for proton transfer (PT), and a correct description of its properties is essential in order to understand the PT mechanism in more complex aqueous systems. Our Jastrow correlated AGP wave function ensures an accurate treatment of electron correlation. By exploiting the advantage of contracting the primitive basis set over atomic hybrid orbitals, we are able to limit dramatically the number of variational parameters with a systematic control on the numerical precision, a crucial ingredient in order to simulate larger systems. For both energetics and geometrical properties, our QMC results are found to be in excellent agreement with state-of-the-art coupled cluster CCSD(T) techniques. A comparison with density functional theory in the PBE approximation points to the crucial role of electron correlation for a correct description of the PT in the dimer. We prove that the QMC framework used in this work is able to resolve the tiny energy differences (similar to 0.3 kcal/mol) and structural variations involved in proton transfer reactions. Our approach combines these features and a favorable N-4 scaling with the number of particles which paves the way to the simulation of more realistic PT models. A test calculation on a larger protonated water cluster is carried out. The QMC approach used here represents a promising candidate to provide the first high-level ab initio description of PT in water