577 research outputs found
Classical and quantum algorithms for scaling problems
This thesis is concerned with scaling problems, which have a plethora of connections to different areas of mathematics, physics and computer science. Although many structural aspects of these problems are understood by now, we only know how to solve them efficiently in special cases.We give new algorithms for non-commutative scaling problems with complexity guarantees that match the prior state of the art. To this end, we extend the well-known (self-concordance based) interior-point method (IPM) framework to Riemannian manifolds, motivated by its success in the commutative setting. Moreover, the IPM framework does not obviously suffer from the same obstructions to efficiency as previous methods. It also yields the first high-precision algorithms for other natural geometric problems in non-positive curvature.For the (commutative) problems of matrix scaling and balancing, we show that quantum algorithms can outperform the (already very efficient) state-of-the-art classical algorithms. Their time complexity can be sublinear in the input size; in certain parameter regimes they are also optimal, whereas in others we show no quantum speedup over the classical methods is possible. Along the way, we provide improvements over the long-standing state of the art for searching for all marked elements in a list, and computing the sum of a list of numbers.We identify a new application in the context of tensor networks for quantum many-body physics. We define a computable canonical form for uniform projected entangled pair states (as the solution to a scaling problem), circumventing previously known undecidability results. We also show, by characterizing the invariant polynomials, that the canonical form is determined by evaluating the tensor network contractions on networks of bounded size
On the structure of repeated-root polycyclic codes over local rings
ProducciĂłn CientĂficaThis paper provides the Generalized Mattson Solomon polynomial for repeated-root polycyclic codes over local rings that gives an explicit decomposition of them in terms of idempotents. It also states some structural properties of repeated-root polycyclic codes over finite fields in terms of matrix product codes. Both approaches provide a description of the -dual code for a given polycyclic code.MCIN/AEI /10.13039/501100011033 - EU NextGenerationEU/ PRTR (Grant TED2021-130358B-I00)Bulgarian Ministry of Education and Science, Scientific Programme âEnhancing the Research Capacity in Mathematical Sciences (PIKOM)â, No. DO1-67/05.05.2022.TĂBËITAK within the scope of 2219 International Post Doctoral Research Fellowship Program with application number 1059B19210116
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Algebraic construction of the sigma function for general Weierstrass curves
The Weierstrass curve is a smooth algebraic curve determined by the
Weierstrass canonical form, , where is a positive integer, and each
is a polynomial in with a certain degree. It is known that every compact
Riemann surface has a Weierstrass curve which is birational to the surface.
The form provides the projection as a covering
space. Let and
. Recently we have the explicit
description of the complementary module of
-module , which leads the explicit expressions of the
holomorphic one form except , and the trace operator such that
for for . In terms of them, we express the fundamental 2-form of
the second kind and a connection to the sigma functions for .Comment: 34pages. arXiv admin note: substantial text overlap with
arXiv:2207.0190
Exactly soluble models in many-body physics
Almost all phenomena in the universe are described, at the fundamental level, by quantum manybody
models. In general, however, a complete understanding of large systems with many degrees of
freedom is impossible. While in general many-body quantum systems are intractable, there are
special cases for which there are techniques that allow for an exact solution.
Exactly soluble models are interesting because they are soluble; beyond this, they can be used to
gain intuition for further reaching many-body systems, including when they can be leveraged to help
with numerical approximations for general models. The work presented in this thesis considers
exactly soluble models of quantum many-body systems.
The first part of this thesis extends the family of many-body spin models for which we can find a freefermion
solution.
A solution method that was developed for a specific free-fermion model is generalized in such a way
that allows application to a broader class of many-body spin system than was previously known to be
free. Models which admit a solution via this method are characterized by a graph theory invariants: in
brief it is shown that a quantum spin system has an exact description via non-interacting fermions if
its frustration graph is claw-free and contains a simplicial clique.
The second part of this thesis gives an explicit example of how the usefulness of exactly soluble
models can extend beyond the solution itself. This chapter pertains to the calculation of the
topological entanglement entropy in topologically ordered loop-gas states. Topological entanglement
entropy gives an understanding of how correlations may extend throughout a system. In this chapter
the topological entanglement entropy of two- and three-dimensional loop-gas states is calculated in
the bulk and at the boundary. We obtain a closed form expression for the topological entanglement in
terms of the anyonic theory that the models support
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
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
Undergraduate and Graduate Course Descriptions, 2023 Spring
Wright State University undergraduate and graduate course descriptions from Spring 2023
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