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Learning With Errors and Extrapolated Dihedral Cosets [article]

Zvika Brakerski and Elena Kirshanova and Damien Stehlé and Weiqiang Wen
2019 arXiv   pre-print
The hardness of the learning with errors (LWE) problem is one of the most fruitful resources of modern cryptography.  ...  We show that under quantum polynomial time reductions, LWE is equivalent to a relaxed version of the dihedral coset problem (DCP), which we call extrapolated DCP (eDCP).  ...  Analogous to LWE, it has two versions: search and decisional. Definition 4 (Search Extrapolated Dihedral Coset Problem).  ... 
arXiv:1710.08223v2 fatcat:pyjatz5n3rdbtn4vcgkzmmkgxe

Learning with Errors and Extrapolated Dihedral Cosets [chapter]

Zvika Brakerski, Elena Kirshanova, Damien Stehlé, Weiqiang Wen
2018 Lecture Notes in Computer Science  
The hardness of the learning with errors (LWE) problem is one of the most fruitful resources of modern cryptography.  ...  We show that under quantum polynomial time reductions, LWE is equivalent to a relaxed version of the dihedral coset problem (DCP), which we call extrapolated DCP (eDCP).  ...  the Extrapolated Dihedral Coset problems (lower-left).  ... 
doi:10.1007/978-3-319-76581-5_24 fatcat:7pnqjrajbfcyro4zqsc3dqwpcu

Quantum algorithms for typical hard problems: a perspective of cryptanalysis

Jingwen Suo, Licheng Wang, Sijia Yang, Wenjie Zheng, Jiankang Zhang
2020 Quantum Information Processing  
problem, discrete logarithmic problem and its variants, lattice problem, dihedral hidden subgroup problems and extrapolated dihedral coset problem.  ...  With the purpose to bridge the two disciplines, it is significant to summarize known quantum algorithms and their threats toward these cryptographic intractable problems from a perspective of cryptanalysis  ...  as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.  ... 
doi:10.1007/s11128-020-02673-x fatcat:xwzsxob33baftd27y555sihd5a

Quantum Algorithms for Variants of Average-Case Lattice Problems via Filtering [article]

Yilei Chen and Qipeng Liu and Mark Zhandry
2021 arXiv   pre-print
, including bounded uniform distributions and Laplace distributions. (*) Extrapolated dihedral coset problem (EDCP) with certain parameters.  ...  the bound of infinity norm is set to be half of the modulus minus a constant. (*) Learning with errors (LWE) problem given LWE-like quantum states with polynomially large moduli and certain error distributions  ...  We would also like to thank Luowen Qian, Léo Ducas, and the anonymous reviewers for their helpful comments.  ... 
arXiv:2108.11015v2 fatcat:ccepblqopvhzpmrtwdkqngbile

Abelian Hypergroups and Quantum Computation [article]

Juan Bermejo-Vega, Kevin C. Zatloukal
2015 arXiv   pre-print
we develop a stabilizer formalism using abelian hypergroups and an associated classical simulation theorem (a la Gottesman-Knill).  ...  We also give efficient methods for manipulating non-unitary, non-monomial stabilizers and an adaptive Fourier sampling technique of general interest.  ...  Jozsa, for providing references; and the MIT Center for Theoretical Physics and the Max Planck Institute of Quantum Optics for hosting us in 2013 and 2014.  ... 
arXiv:1509.05806v2 fatcat:wyxhjpxzrbfiza34nohr4p4fxa

Logical Randomized Benchmarking [article]

Joshua Combes, Christopher Granade, Christopher Ferrie, Steven T. Flammia
2017 arXiv   pre-print
Extrapolating physical error rates to logical error rates requires many assumptions and thus can radically under- or overestimate the performance of an error correction implementation.  ...  We show that our method reliably reports logical performance and can estimate the average probability of correctable and uncorrectable errors for a given code and physical channel.  ...  of Research, Innovation and Science.  ... 
arXiv:1702.03688v1 fatcat:53lygjngsnbe3dfrikfhdzhk3q

Two quantum Ising algorithms for the Shortest Vector Problem: one for now and one for later [article]

David Joseph, Adam Callison, Cong Ling, Florian Mintert
2021 arXiv   pre-print
New cryptosystems are being designed and standardised for the post-quantum era, and a significant proportion of these rely on the hardness of problems like the Shortest Vector Problem to a quantum adversary  ...  Analysis of the algorithms' performance on a quantum annealer and in numerical simulations show that the more qubit-efficient variant will outperform in the long run, while the other variant is more suitable  ...  ACKNOWLEDGEMENTS The authors would like to thank Charles Grover for his helpful discussions on the proof of Theorem III.1, and D-Wave for providing Fig 7.  ... 
arXiv:2006.14057v7 fatcat:xmom26i6lzbsppweywvgrwf2yy

Normalizer Circuits and Quantum Computation [article]

Juan Bermejo-Vega
2016 arXiv   pre-print
This thesis also establishes a precise connection between Shor's quantum algorithm and the stabilizer formalism, revealing a common mathematical structure in several quantum speed-ups and error-correcting  ...  continuous-variable systems, hybrid settings, and anyonic systems.  ...  and error-correcting codes.  ... 
arXiv:1611.09274v1 fatcat:y6tw3ta2mfb5fkpzdyrbrmuhwu

Quantum Cryptanalysis (Dagstuhl Seminar 19421)

Michele Mosca, Maria Naya-Plasencia, Rainer Steinwandt, Michael Wagner
2020 Dagstuhl Reports  
Different quantum attack models can be considered here, and attack models that are close to being realizable with today's technology are particularly relevant.  ...  → Computational complexity and cryptography License Creative Commons BY 3.0 Unported license © Michele Mosca, María Naya-Plasencia and Rainer Steinwandt Motivation and scope This fifth installment of a  ...  This algorithm relies on the Sieve of Kuperberg which solves the Dihedral Coset Problem. We insist on a recent result showing that we can trade off quantum effort for classical one.  ... 
doi:10.4230/dagrep.9.10.47 dblp:journals/dagstuhl-reports/MoscaNS19 fatcat:b4fbhk267zhsdch2lotalrzwoa

Survey of Promising Technologies for Quantum Drones and Networks

Adarsh Kumar, Surbhi Bhatia, Keshav Kaushik, Manjula Gandhi, Gayathri Devi, Diego Pacheco, Arwa Mashat
2021 IEEE Access  
challenge, and its variants, extrapolated dihedral coset problem, lattice problem, hidden subgroup problems, and dihedral problem, were discussed in [5] .  ...  mitigation, (Error extrapolation and quasi probability decomposition), E: Other methods Figure 5 : 5 Global quantum communications network.  ... 
doi:10.1109/access.2021.3109816 fatcat:gkyevfoqrfertef7j7gjzcstii

Valence-bond solids, vestigial order, and emergent SO(5) symmetry in a two-dimensional quantum magnet [article]

Jun Takahashi, Anders W. Sandvik
2020 arXiv   pre-print
We carry out a detailed quantum Monte Carlo study of the quantum phase transition between the antiferromagnetic (AFM) and PVBS states.  ...  We find a first-order transition, in contrast to previously studied continuous 'deconfined' transitions between the AFM and columnar valence-bond solids.  ...  Fig. 5 extrapolates to a finite value.  ... 
arXiv:2001.10045v1 fatcat:clhii2vwanellp2fpmkhssktqq

Variables separated equations: Strikingly different roles for the Branch Cycle Lemma and the Finite Simple Group Classification [article]

Michael d. Fried
2011 arXiv   pre-print
with Chow motive coefficients.  ...  Davenport's Problem asks: What can we expect of two polynomials, over the integers, with the same ranges on almost all residue class fields?  ...  Müller says [Fe73] was what he first saw of the details of Davenport pairs, and he corrected an error in that.  ... 
arXiv:1012.5297v5 fatcat:r5vhhbehd5agdbsznvpgufgkpm

A Differential Topological Model for Olfactory Learning and Representation [article]

Jack A. Cook
2020 arXiv   pre-print
This thesis is designed to be a self-contained exposition of the neurobiological and mathematical aspects of sensory perception, memory, and learning with a bias towards olfaction.  ...  The method of generalization is extrapolate information from one stimulus and use this to "learn" something about its nearest neighbors in the perceptual space.  ...  Just as with the dihedral groups D n , we can ask how a group may act on a set; that is, how does it permute the elements?  ... 
arXiv:2009.02310v1 fatcat:yzx7qhpi5fgvhigj6wqqzw4e2i

Monte Carlo computations in lattice gauge theories

Michael Creutz, Laurence Jacobs, Claudio Rebbi
1983 Physics reports  
extrapolations.  ...  value but is affected by larger errors).  ... 
doi:10.1016/0370-1573(83)90016-9 fatcat:znzsr2igm5gmhcxkpz6ec3lt3q

Categorical Quantum Dynamics [article]

Stefano Gogioso
2017 arXiv   pre-print
duality, the connection of strong complementarity with the Weyl Canonical Commutation Relations, the generalisations of Feynman's clock construction, the existence of time observables and the emergence  ...  All in all, we argue that strong complementarity is a truly powerful and versatile building block for quantum theory and its applications, and one that should draw a lot more attention in the future.  ...  stating that, relative to oracles with the promise above, Simon's problem separates BQP (the class of bounded-error quantum polynomial time problems) from BPP (the class of bounded-error classical polynomial  ... 
arXiv:1709.09772v1 fatcat:idax6g6zkfdwdksdh3zwea3mpa
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