Spherical and Hyperbolic Toric Topology-Based Codes On Graph Embedding for Ising MRF Models: Classical and Quantum Topology Machine Learning

The paper introduces the application of information geometry to describe the ground states of Ising models by utilizing parity-check matrices of cyclic and quasi-cyclic codes on toric and spherical topologies. The approach establishes a connection between machine learning and error-correcting coding. This proposed approach has implications for the development of new embedding methods based on trapping sets. Statistical physics and number geometry applied for optimize error-correcting codes, leading to these embedding and sparse factorization methods. The paper establishes a direct connection between DNN architecture and error-correcting coding by demonstrating how state-of-the-art architectures (ChordMixer, Mega, Mega-chunk, CDIL, ...) from the long-range arena can be equivalent to of block and convolutional LDPC codes (Cage-graph, Repeat Accumulate). QC codes correspond to certain types of chemical elements, with the carbon element being represented by the mixed automorphism Shu-Lin-Fossorier QC-LDPC code. The connections between Belief Propagation and the Permanent, Bethe-Permanent, Nishimori Temperature, and Bethe-Hessian Matrix are elaborated upon in detail. The Quantum Approximate Optimization Algorithm (QAOA) used in the Sherrington-Kirkpatrick Ising model can be seen as analogous to the back-propagation loss function landscape in training DNNs. This similarity creates a comparable problem with TS pseudo-codeword, resembling the belief propagation method. Additionally, the layer depth in QAOA correlates to the number of decoding belief propagation iterations in the Wiberg decoding tree. Overall, this work has the potential to advance multiple fields, from Information Theory, DNN architecture design (sparse and structured prior graph topology), efficient hardware design for Quantum and Classical DPU/TPU (graph, quantize and shift register architect.) to Materials Science and beyond.Comment: 71 pages, 42 Figures, 1 Table, 1 Appendix. arXiv admin note: text overlap with arXiv:2109.08184 by other author

Shapelets for gravitational lensing and galaxy morphology studies

The presented work is concerned with the morphological description of stars and galaxies in the framework of the shapelet method. This method constitutes a linear expansion in the orthonormal set of Gauss-Hermite polynomials. Its main advantages – linearity, compactness, invariance under Fourier transformation, and the relation to the moments of the brightness distribution – are extensively discussed. The practical treatment of the image decomposition and of the deconvolution from the point spread function are further elaborated. Moreover, three fields of application are presented together with new investigations on the applicability and validity of the method: weak gravitational lensing, morphological class discovery, and realistic simulation of extragalactic observations

Coding for Communications and Secrecy

Shannon, in his landmark 1948 paper, developed a framework for characterizing the fundamental limits of information transmission. Among other results, he showed that reliable communication over a channel is possible at any rate below its capacity. In 2008, Arikan discovered polar codes; the only class of explicitly constructed low-complexity codes that achieve the capacity of any binary-input memoryless symmetric-output channel. Arikan's polar transform turns independent copies of a noisy channel into a collection of synthetic almost-noiseless and almost-useless channels. Polar codes are realized by sending data bits over the almost-noiseless channels and recovering them by using a low-complexity successive-cancellation (SC) decoder, at the receiver. In the first part of this thesis, we study polar codes for communications. When the underlying channel is an erasure channel, we show that almost all correlation coefficients between the erasure events of the synthetic channels decay rapidly. Hence, the sum of the erasure probabilities of the information-carrying channels is a tight estimate of the block-error probability of polar codes when used for communication over the erasure channel. We study SC list (SCL) decoding, a method for boosting the performance of short polar codes. We prove that the method has a numerically stable formulation in log-likelihood ratios. In hardware, this formulation increases the decoding throughput by 53% and reduces the decoder's size about 33%. We present empirical results on the trade-off between the length of the CRC and the performance gains in a CRC-aided version of the list decoder. We also make numerical comparisons of the performance of long polar codes under SC decoding with that of short polar codes under SCL decoding. Shannon's framework also quantifies the secrecy of communications. Wyner, in 1975, proposed a model for communications in the presence of an eavesdropper. It was shown that, at rates below the secrecy capacity, there exist reliable communication schemes in which the amount of information leaked to the eavesdropper decays exponentially in the block-length of the code. In the second part of this thesis, we study the rate of this decay. We derive the exact exponential decay rate of the ensemble-average of the information leaked to the eavesdropper in Wyner's model when a randomly constructed code is used for secure communications. For codes sampled from the ensemble of i.i.d. random codes, we show that the previously known lower bound to the exponent is exact. Our ensemble-optimal exponent for random constant-composition codes improves the lower bound extant in the literature. Finally, we show that random linear codes have the same secrecy power as i.i.d. random codes. The key to securing messages against an eavesdropper is to exploit the randomness of her communication channel so that the statistics of her observation resembles that of a pure noise process for any sent message. We study the effect of feedback on this approximation and show that it does not reduce the minimum entropy rate required to approximate a given process. However, we give examples where variable-length schemes achieve much larger exponents in this approximation in the presence of feedback than the exponents in systems without feedback. Upper-bounding the best exponent that block codes attain, we conclude that variable-length coding is necessary for achieving the improved exponents

Analysis and design of physical-layer network coding for relay networks

Physical-layer network coding (PNC) is a technique to make use of interference in wireless transmissions to boost the system throughput. In a PNC employed relay network, the relay node directly recovers and transmits a linear combination of its received messages in the physical layer. It has been shown that PNC can achieve near information-capacity rates. PNC is a new information exchange scheme introduced in wireless transmission. In practice, transmitters and receivers need to be designed and optimized, to achieve fast and reliable information exchange. Thus, we would like to ask: How to design the PNC schemes to achieve fast and reliable information exchange? In this thesis, we address this question from the following works: Firstly, we studied channel-uncoded PNC in two-way relay fading channels with QPSK modulation. The computation error probability for computing network coded messages at the relay is derived. We then optimized the network coding functions at the relay to improve the error rate performance. We then worked on channel coded PNC. The codes we studied include classical binary code, modern codes, and lattice codes. We analyzed the distance spectra of channel-coded PNC schemes with classical binary codes, to derive upper bounds for error rates of computing network coded messages at the relay. We designed and optimized irregular repeat-accumulate coded PNC. We modified the conventional extrinsic information transfer chart in the optimization process to suit the superimposed signal received at the relay. We analyzed and designed Eisenstein integer based lattice coded PNC in multi-way relay fading channels, to derive error rate performance bounds of computing network coded messages. Finally we extended our work to multi-way relay channels. We proposed a opportunistic transmission scheme for a pair-wise transmission PNC in a single-input single-output multi-way relay channel, to improve the sum-rate at the relay. The error performance of computing network coded messages at the relay is also improved. We optimized the uplink/downlink channel usage for multi-input multi-output multi-way relay channels with PNC to maximize the degrees of freedom capacity. We also showed that the system sum-rate can be further improved by a proposed iterative optimization algorithm

Fault-tolerance in two-dimensional topological systems

This thesis is a collection of ideas with the general goal of building, at least in the abstract, a local fault-tolerant quantum computer. The connection between quantum information and topology has proven to be an active area of research in several fields. The introduction of the toric code by Alexei Kitaev demonstrated the usefulness of topology for quantum memory and quantum computation. Many quantum codes used for quantum memory are modeled by spin systems on a lattice, with operators that extract syndrome information placed on vertices or faces of the lattice. It is natural to wonder whether the useful codes in such systems can be classified. This thesis presents work that leverages ideas from topology and graph theory to explore the space of such codes. Homological stabilizer codes are introduced and it is shown that, under a set of reasonable assumptions, any qubit homological stabilizer code is equivalent to either a toric code or a color code. Additionally, the toric code and the color code correspond to distinct classes of graphs. Many systems have been proposed as candidate quantum computers. It is very desirable to design quantum computing architectures with two-dimensional layouts and low complexity in parity-checking circuitry. Kitaev\u27s surface codes provided the first example of codes satisfying this property. They provided a new route to fault tolerance with more modest overheads and thresholds approaching 1%. The recently discovered color codes share many properties with the surface codes, such as the ability to perform syndrome extraction locally in two dimensions. Some families of color codes admit a transversal implementation of the entire Clifford group. This work investigates color codes on the 4.8.8 lattice known as triangular codes. I develop a fault-tolerant error-correction strategy for these codes in which repeated syndrome measurements on this lattice generate a three-dimensional space-time combinatorial structure. I then develop an integer program that analyzes this structure and determines the most likely set of errors consistent with the observed syndrome values. I implement this integer program to find the threshold for depolarizing noise on small versions of these triangular codes. Because the threshold for magic-state distillation is likely to be higher than this value and because logical CNOT gates can be performed by code deformation in a single block instead of between pairs of blocks, the threshold for fault-tolerant quantum memory for these codes is also the threshold for fault-tolerant quantum computation with them. Since the advent of a threshold theorem for quantum computers much has been improved upon. Thresholds have increased, architectures have become more local, and gate sets have been simplified. The overhead for magic-state distillation has been studied, but not nearly to the extent of the aforementioned topics. A method for greatly reducing this overhead, known as reusable magic states, is studied here. While examples of reusable magic states exist for Clifford gates, I give strong reasons to believe they do not exist for non-Clifford gates

Investigations in integrative and molecular bioscience

Modern biology is going through a revolution of new methods and insights resulting from the new availability of high-throughput DNA sequencing technology. I here present work contributing mathematical and computational methods for gaining insight from large DNA sequencing data sets at three distinct levels. First, I present a method for improving the accuracy and efficiency of DNA barcodes, short sequences of DNA used to label individual molecules in pooled samples. Many DNA sequencing applications depend on the use of DNA barcodes. However, errors in DNA synthesis and sequencing—substitutions, insertions, and deletions—confound the correct interpretation of these barcodes. I here present Filled/truncated Right End Edit (FREE) barcodes designed for barcode error-correction in the context of a downstream sequence. Second, I present the Chip-Hybridized Affinity Mapping Platform (CHAMP), a novel technology for repurposing used DNA sequencing chips to study the mechanism and sequence preferences of DNA-binding proteins. Since 2012, the CRISPR family of proteins have gained wide application for their efficiency and ease of use in editing genomes in vivo. Using CHAMP, I, in collaboration with experimentalists in Ilya Finkelstein’s lab, investigated the mechanism and sequence preference of the CRISPR Cascade complex, and discovered a novel periodic lack of sequence specificity in DNA binding. I further determined specific nucleotides important for recruitment of and processing by the nuclease domain, Cas3. Third, I present a meta-analysis of the order Chiroptera, the order of bats, using the new wealth of DNA sequence information of eighteen bat species. The transcriptome sequencing data for two of these bats—Hypsignathus monstrosus and Rousettus aegyptiacus, bats associated with studies of the Ebola and Marburg viruses respectively— is novel to this study. Using all this DNA sequence information, I reconstructed a high- confidence Chiropteran phylogeny and found 299 genes with signatures of positive selection, a signature associated with viral antagonism. Further study of these genes may shed light on the mechanism through which several bat viruses relevant to human health hijack the cell, including SARS, Ebola, Hendra, and NipahComputational Science, Engineering, and Mathematic