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We propose a new strategy to decode color codes, which is based on the projection of the error onto three surface codes. This provides a method to transform every decoding algorithm of surface codes into a decoding algorithm of color codes.…
Advances in hardware and language model architecture have spurred a revolution in natural language generation. However, autoregressive models compute probability distributions over next-token choices, and sampling from these distributions,…
Efficient and accurate decoding of quantum error-correcting codes is essential for fault-tolerant quantum computation, however, it is challenging due to the degeneracy of errors, the complex code topology, and the large space for logical…
We study the channel coding problem when errors and uncertainty occur in the encoding process. For simplicity we assume the channel between the encoder and the decoder is perfect. Focusing on linear block codes, we model the encoding…
I will show that there is a deep relation between error-correction codes and certain mathematical models of spin glasses. In particular minimum error probability decoding is equivalent to finding the ground state of the corresponding spin…
Dynamic Mode Decomposition (DMD) is a data-driven decomposition technique extracting spatio-temporal patterns of time-dependent phenomena. In this paper, we perform a comprehensive theoretical analysis of various variants of DMD. We provide…
We formulate code concatenation as the action of a unitary quantum circuit on an expanding tree geometry and find that for certain classes of gates, applied identically at each node, a binary tree circuit encodes a single logical qubit with…
Time-varying stochastic optimization problems frequently arise in machine learning practice (e.g. gradual domain shift, object tracking, strategic classification). Although most problems are solved in discrete time, the underlying process…
Color coding is an algorithmic technique used in parameterized complexity theory to detect "small" structures inside graphs. The idea is to derandomize algorithms that first randomly color a graph and then search for an easily-detectable,…
A large class of MDS linear codes is constructed. These codes are endowed with an efficient decoding algorithm. Both the definition of the codes and the design of their decoding algorithm only require from Linear Algebra methods, making…
We introduce a convergent iterative algorithm for finding the optimal coding and decoding operations for an arbitrary noisy quantum channel. This algorithm does not require any error syndrome to be corrected completely, and hence also finds…
Hinging on ideas from physical-layer network coding, some promising proposals of coded random access systems seek to improve system performance (while preserving low complexity) by means of packet repetitions and decoding of linear…
In a physical system, changing parameters such as temperature can induce a phase transition: an abrupt change from one state of matter to another. Analogous phenomena have recently been observed in large language models. Typically, the task…
Gallager codes are the best error-correcting codes to-date. In this paper we study them by using the tools of statistical mechanics. The corresponding statistical mechanics model is a spin model on a sparse random graph. The model can be…
Deep generative models for graphs have exhibited promising performance in ever-increasing domains such as design of molecules (i.e, graph of atoms) and structure prediction of proteins (i.e., graph of amino acids). Existing work typically…
Iterative phase retrieval algorithms are widely used in digital optics for their efficiency and simplicity. Conventionally, these algorithms do not consider aberrations as they assume an ideal, aberration-free optical system. Here, we…
Decoding of convolutional codes poses a significant challenge for coding theory. Classical methods, based on e.g. Viterbi decoding, suffer from being computationally expensive and are restricted therefore to codes of small complexity. Based…
Dynamical sampling deals with signals that evolve in time under the action of a linear operator. The purpose of the present paper is to analyze the performance of the basic dynamical sampling algorithms in the finite dimensional case and…
Staircase codes play an important role as error-correcting codes in optical communications. In this paper, a low-complexity method for resolving stall patterns when decoding staircase codes is described. Stall patterns are the dominating…
The gauge theory of spin glasses and statistical-mechanical formulation of error-correcting codes are reviewed with an emphasis on their similarities. For the gauge theory, we explain the functional identities on dynamical autocorrelation…