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Introduction to Haar Measure Tools in Quantum Information: A Beginner's Tutorial

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arXiv 2023年

作者： Mele, Antonio Anna Dahlem Center for Complex Quantum Systems Freie Universität Berlin Berlin14195 Germany

The Haar measure plays a vital role in quantum information, but its study often requires a deep understanding of representation theory, posing a challenge for beginners. This tutorial aims to provide a basic introduction to Haar measure tools in quantum information, utilizing only basic knowledge of linear algebra and thus aiming to make this topic more accessible. The tutorial begins by introducing the Haar measure with a specific emphasis on characterizing the moment operator, an essential element for computing integrals over the Haar measure. It also covers properties of the symmetric subspace and introduces helpful tools like tensor network diagrammatic notation, which aid in visualizing and simplifying calculations. Next, the tutorial explores the concept of unitary designs, providing equivalent definitions, and subsequently explores approximate notions of unitary designs, shedding light on the relationships between these different notions. Practical examples of Haar measure calculations are illustrated, including the derivation of well-known formulas such as the twirling of a quantum channel. Lastly, the tutorial showcases the applications of Haar measure calculations in quantum machine learning and classical shadow tomography. © 2023, CC BY.

关键词： computation theory

Equilibrium Convergence in Large Games

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SSRN

SSRN 2023年

作者： Chen, Enxian Wu, Bin Xu, Hanping School of Economics Nankai University Tianjin300071 China International School of Economics and Management Capital University of Economics and Business Fengtai District Beijing100070 China Department of Mathematics National University of Singapore 10 Lower Kent Ridge Road 119076 China

This paper presents a general closed graph property for (randomized strategy) Nash equilibrium correspondence in large games. In particular, we show that for any large game with a convergent sequence of finite-player games, the limit of any convergent sequence of Nash equilibria of the corresponding finite-player games can be induced by a Nash equilibrium of the large game. Such a result goes beyond earlier results on the closed graph property for pure strategy Nash equilibrium correspondence in large games in multiple aspects and also unifies those earlier results under relevant conditions. © 2023, The Authors. All rights reserved.

关键词： computation theory

Symmetry-resolved modular correlation functions in free fermionic theories

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arXiv 2023年

作者： Giulio, Giuseppe Di Erdmenger, Johanna Institute for Theoretical Physics and Astrophysics and Würzburg-Dresden Cluster of Excellence ct.qmat Julius-Maximilians-Universität Würzburg Am Hubland Würzburg97074 Germany

As a new ingredient for analyzing the fine structure of entanglement, we study the symmetry resolution of the modular flow of U(1)-invariant operators in theories endowed with a global U(1) symmetry. We provide a consistent definition of symmetry-resolved modular flow that is defined for a local algebra of operators associated to a sector with fixed charge. We also discuss the symmetry-resolved modular correlation functions and show that they satisfy the KMS condition in each symmetry sector. Our analysis relies on the factorization of the Hilbert space associated to spatial subsystems. We provide a toolkit for computing the symmetry-resolved modular correlation function of the charge density operator in free fermionic theories. As an application, we compute this correlation function for a 1 + 1-dimensional free massless Dirac field theory and find that it is independent of the charge sector at leading order in the ultraviolet cutoff expansion. This feature can be regarded as a charge equipartition of the modular correlation function. Although obtained for free fermions, these results may be of potential interest for bulk reconstruction in AdS/CFT. Copyright © 2023, The Authors. All rights reserved.

关键词： computation theory

Electromagnetic fields in compact binaries: post-Newtonian wave generation and application to double white dwarfs systems

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arXiv 2023年

作者： Henry, Quentin Larrouturou, François Le Poncin-Lafitte, Christophe PotsdamD-14476 Germany Deutsches Elektronen-Synchrotron DESY Notkestr. 85 Hamburg22607 Germany SYRTE Observatoire de Paris Université PSL CNRS Sorbonne Université LNE 61 avenue de l’Observatoire Paris75014 France

The aim of this work is twofold: (i) to properly define a wave-generation formalism for compact-supported sources embedded in Einstein-Maxwell theory, relying on matched post-Newtonian and multipolar-post-Minkowskian expansions;(ii) to apply this formalism (which is valid for any type of post-Newtonian sources) to the case of two stars with constant and aligned magnetic dipoles, by computing the fluxes of energy and angular momentum to the next-to-leading order, as well as the gravitational amplitude modes. Assuming eccentric orbits, we derive the evolution of orbital parameters, as well as the observables of the system, notably the gravitational phase for quasicircular orbits. Finally, we give some numerical estimates for the contribution of the magnetic dipoles for some realistic systems. © 2023, CC BY.

关键词： computation theory

Non-Pauli errors can be efficiently sampled in qudit surface codes

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arXiv 2023年

作者： Ma, Yue Hanks, Michael Kim, M.S. QOLS Blackett Laboratory Imperial College London LondonSW7 2AZ United Kingdom

Surface codes are the most promising candidates for fault-tolerant quantum computation. Single qudit errors are typically modelled as Pauli operators, to which general errors are converted via randomizing methods. In this Letter, we quantify remaining correlations after syndrome measurement for a qudit 2D surface code subject to non-Pauli errors. Using belief propagation and percolation theory, we relate correlations to loops on the lattice. Below the error correction threshold, remaining correlations are sparse and locally constrained. Syndromes for qudit surface codes are therefore efficiently samplable for non-Pauli errors, independent of the exact forms of the error and decoder. Copyright © 2023, The Authors. All rights reserved.

关键词： computation theory

Computing nonsurjective primes associated to Galois representations of genus 2 curves

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arXiv 2023年

作者： Banwait, Barinder S. Brumer, Armand Kim, Hyun Jong Klagsbrun, Zev Mayle, Jacob Srinivasan, Padmavathi Vogt, Isabel Department of Mathematics & Statistics Boston University BostonMA United States Department of Mathematics Fordham University New YorkNY United States Department of Mathematics University of Wisconsin-Madison MadisonWI United States Center for Communications Research San DiegoCA United States Department of Mathematics Wake Forest University Winston-SalemNC United States ICERM ProvidenceRI United States Department of Mathematics Brown University ProvidenceRI United States

For a genus 2 curve C over Q whose Jacobian A admits only trivial geometric endomorphisms, Serre’s open image theorem for abelian surfaces asserts that there are only finitely many primes ` for which the Galois action on `-torsion points of A is not maximal. Building on work of Dieulefait, we give a practical algorithm to compute this finite set. The key inputs are Mitchell’s classification of maximal subgroups of PSp4(F`), sampling of the characteristic polynomials of Frobenius, and the Khare–Wintenberger modularity theorem. The algorithm has been submitted for integration into Sage, executed on all of the genus 2 curves with trivial endomorphism ring in the LMFDB, and the results incorporated into the homepage of each such *** Codes 11F80(primary), 11G10, 11Y16 © 2023, CC BY.

关键词： computation theory

A coparative study of models of parallel coputation 15

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A coparative study of models of parallel coputation

15th Annual Symposium on Switching and Automata theory, SWAT 1974

作者： Lipton, R.J. Snyder, L. Zalcstein, Y. Computer Science Department Yale University New HavenCT06520 United States Computer Science Department State University of New York Stony BrookNY11790 United States

In recent years, a large number of models of parallel computation have been proposed. These models have rather diverse motivations including problems from operating systems, hardware, numerical algorithms and others. In a recent thesis, Petersonl has proved that most of these models are equivalent in the sense that they generate the same set of "computations." This notion of equivalence, which is a straightforward generalization of the notion of equivalence of gramrs in automata theory, is rather unsatisfactory in that it does not distinguish among models that intuitively are not equivalent. © 2021 IEEE. All rights reserved.

关键词： computation theory

computation of Gabor coefficients for objects with polygonal cross section 22

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Computation of Gabor coefficients for objects with polygonal...

22nd International Conference on Electromagnetics in Advanced Applications, ICEAA 2021

作者： Sun, L. Eijsvogel, S. Sepehripour, F. Dilz, R.J. Van Beurden, M.C. Eindhoven University of Technology Department of Electrical Engineering Eindhoven Netherlands

ISBN: (纸本)9781665413862

The Gabor transformation connects the spectral and spatial domain in a recently developed spatial spectral Maxwell solver. A key step involves computing the Gabor coefficients of the characteristic function of dielectric scattering objects. Therefore, computing Gabor coefficients accurately and efficiently is significant in the further development of this Maxwell solver. We discuss a method to numerically calculate the integrals involved in computing the Gabor coefficients, based on Gauss's theorem and recurrence relations. © 2021 IEEE.

关键词： computation theory

Towards Mesh-Free Patient-Specific Mitral Valve Modeling 11th

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Towards Mesh-Free Patient-Specific Mitral Valve Modeling

11th International Workshop on Statistical Atlases and computational Models of the Heart, STACOM 2020 held in Conjunction with MICCAI 2020

作者： Ros, Judit Camara, Oscar Hermida, Uxio Bijnens, Bart Morales, Hernán G. Physense BCN MedTech Department of Information and Communication Technologies Universitat Pompeu Fabra Barcelona Spain Philips Research Paris Suresnes France ICREA Barcelona Spain

ISBN: (纸本)9783030681067

computational modeling is a tool that has gained importance recently to better understand valve physiopathology, to assess safety and efficacy of cardiovascular devices and as a supporting tool for therapy planning. Mesh-based methods, such as the Finite Element Method (FEM), have shown high accuracy and application modeling the mitral valve (MV). However, when it comes to irregular and complex geometries, FEM techniques suffer from well-documented limitations, such as the labor and time-consuming intrinsic need of a mesh. In this work, novel structural models of the MV and mitral valve regurgitation (MVR) dynamics are presented using a mesh-free method. Obtained results show that the developed models are capable of reproducing MV and MVR behaviour with good agreement with respect to both in-vivo and in-silico studies, in terms of valve closure and opening, valve deformation, as well as stress magnitudes. This paper shows that mesh-free methods have the potential to become a powerful alternative to the currently most used modeling approaches. © 2021, Springer Nature Switzerland AG.

关键词： computation theory