Instanton-based Techniques for Analysis and Reduction of Error Floors of LDPC Codes

We describe a family of instanton-based optimization methods developed recently for the analysis of the error floors of low-density parity-check (LDPC) codes. Instantons are the most probable configurations of the channel noise which result in decoding failures. We show that the general idea and the respective optimization technique are applicable broadly to a variety of channels, discrete or continuous, and variety of sub-optimal decoders. Specifically, we consider: iterative belief propagation (BP) decoders, Gallager type decoders, and linear programming (LP) decoders performing over the additive white Gaussian noise channel (AWGNC) and the binary symmetric channel (BSC). The instanton analysis suggests that the underlying topological structures of the most probable instanton of the same code but different channels and decoders are related to each other. Armed with this understanding of the graphical structure of the instanton and its relation to the decoding failures, we suggest a method to construct codes whose Tanner graphs are free of these structures, and thus have less significant error floors.Comment: To appear in IEEE JSAC On Capacity Approaching Codes. 11 Pages and 6 Figure

Nonadiabatic instanton rate theory beyond the golden-rule limit

Fermi's golden rule describes the leading-order behaviour of the reaction rate as a function of the diabatic coupling. Its asymptotic $(\hbar \rightarrow 0)$ limit is the semiclassical golden-rule instanton rate theory, which rigorously approximates nuclear quantum effects, lends itself to efficient numerical computation and gives physical insight into reaction mechanisms. However the golden rule by itself becomes insufficient as the strength of the diabatic coupling increases, so higher-order terms must be additionally considered. In this work we give a first-principles derivation of the next-order term beyond the golden rule, represented as a sum of three components. Two of them lead to new instanton pathways that extend the golden-rule case and, among other factors, account for the effects of recrossing on the full rate. The remaining component derives from the equilibrium partition function and accounts for changes in potential energy around the reactant and product wells due to diabatic coupling. The new semiclassical theory demands little computational effort beyond a golden-rule instanton calculation. It makes it possible to rigorously assess the accuracy of the golden-rule approximation and sets the stage for future work on general semiclassical nonadiabatic rate theories

Inflation with a graceful exit in a random landscape

We develop a stochastic description of small-field inflationary histories with a graceful exit in a random potential whose Hessian is a Gaussian random matrix as a model of the unstructured part of the string landscape. The dynamical evolution in such a random potential from a small-field inflation region towards a viable late-time de Sitter (dS) minimum maps to the dynamics of Dyson Brownian motion describing the relaxation of non-equilibrium eigenvalue spectra in random matrix theory. We analytically compute the relaxation probability in a saddle point approximation of the partition function of the eigenvalue distribution of the Wigner ensemble describing the mass matrices of the critical points. When applied to small-field inflation in the landscape, this leads to an exponentially strong bias against small-field ranges and an upper bound $N\ll 10$ on the number of light fields $N$ participating during inflation from the non-observation of negative spatial curvature.Comment: Published versio

Tunneling and Zero-Point Energy Effects in Multidimensional Hydrogen Transfer Reactions: From Gas Phase to Adsorption on Metal Surfaces

Hydrogen transfer reactions play a significant role in many technological applications and fundamental processes in nature. Despite appearing to be simple reactions, they constitute complex processes where nuclear quantum effects (NQE) such as zero-point energy and nuclear tunneling play a decisive role even at ambient temperature. Moreover, the anharmonic coupling between different degrees of freedom that take place in realistic systems leads to hydrogen dynamics that, in many cases, are hard to interpret and understand. Systematic and quantitative ab initio studies of hydrogen dynamics were performed in systems ranging from gas phase molecules to adsorbates on metallic surfaces using state-of-the-art methodologies based on the path integral formulation of quantum mechanics in combination with the density functional approximation. In order to achieve this task, the construction of a general infrastructure that made the required ring polymer instanton simulations feasible was created, and a new approximation which considerably reduces the computational cost of including NQE on weakly bound systems was proposed and tested in the study of water dissociation at Pt(221) surface. Practical guidelines and limitations were also discussed to help the adoption of such methodologies by the community. The system of choice for most of the studies presented in this thesis was the porphycene molecule, a paradigmatic example of a molecular switch. The are a large number of experimental results in well-controlled environments available in the literature which have demonstrated the importance of NQE and multidimensional coupling for this molecule. Therefore, the porphycene molecule provides the unique possibility to theoretically address these effects and compare the theoretical predictions with experimental results in different environments. A portion of this thesis focuses on the study of porphycene molecule in the gas phase. For this purpose, the intramolecular double hydrogen transfer (DHT) rates and vibrational spectrum were calculated. The theoretical results showed a remarkable agreement with the experiments, and enabled the explanation of the unusual infrared spectra, the elucidation of the dominant DHT mechanism, and the understanding of their temperature dependence. In all the cases, the coupling between low- and high-frequency modes proved to be essential to get qualitatively correct trends. Another portion of this thesis examines molecules adsorbed on surfaces. Studies of porphycene molecules adsorbed on (111) and (110) metal surfaces showed that the stronger the surface-molecule interaction is, the more the molecule buckles upon adsorption, leading to an overall decrease of the DHT rates. The simulations identified different temperature regimes of the DHT mechanism, which was not possible by experimental measurements, and evidenced the importance of surface fluctuations on the DHT rates. In conclusion, this thesis provides a stepping stone towards the understanding of the impact of NQE, anharmonic effects, and multidimensional mode coupling on hydrogen dynamics, and also describes novel computational tools to approach their study by using first-principle calculations