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2021 ◽  
Vol 13 (1) ◽  
pp. 149-159
Author(s):  
T.O. Banakh ◽  
V.M. Gavrylkiv

A subset $B$ of a group $G$ is called a basis of $G$ if each element $g\in G$ can be written as $g=ab$ for some elements $a,b\in B$. The smallest cardinality $|B|$ of a basis $B\subseteq G$ is called the basis size of $G$ and is denoted by $r[G]$. We prove that each finite group $G$ has $r[G]>\sqrt{|G|}$. If $G$ is Abelian, then $r[G]\ge \sqrt{2|G|-|G|/|G_2|}$, where $G_2=\{g\in G:g^{-1} = g\}$. Also we calculate the basis sizes of all Abelian groups of order $\le 60$ and all non-Abelian groups of order $\le 40$.


2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Mario Motta ◽  
Erika Ye ◽  
Jarrod R. McClean ◽  
Zhendong Li ◽  
Austin J. Minnich ◽  
...  

AbstractThe quantum simulation of quantum chemistry is a promising application of quantum computers. However, for N molecular orbitals, the $${\mathcal{O}}({N}^{4})$$ O ( N 4 ) gate complexity of performing Hamiltonian and unitary Coupled Cluster Trotter steps makes simulation based on such primitives challenging. We substantially reduce the gate complexity of such primitives through a two-step low-rank factorization of the Hamiltonian and cluster operator, accompanied by truncation of small terms. Using truncations that incur errors below chemical accuracy allow one to perform Trotter steps of the arbitrary basis electronic structure Hamiltonian with $${\mathcal{O}}({N}^{3})$$ O ( N 3 ) gate complexity in small simulations, which reduces to $${\mathcal{O}}({N}^{2})$$ O ( N 2 ) gate complexity in the asymptotic regime; and unitary Coupled Cluster Trotter steps with $${\mathcal{O}}({N}^{3})$$ O ( N 3 ) gate complexity as a function of increasing basis size for a given molecule. In the case of the Hamiltonian Trotter step, these circuits have $${\mathcal{O}}({N}^{2})$$ O ( N 2 ) depth on a linearly connected array, an improvement over the $${\mathcal{O}}({N}^{3})$$ O ( N 3 ) scaling assuming no truncation. As a practical example, we show that a chemically accurate Hamiltonian Trotter step for a 50 qubit molecular simulation can be carried out in the molecular orbital basis with as few as 4000 layers of parallel nearest-neighbor two-qubit gates, consisting of fewer than 105 non-Clifford rotations. We also apply our algorithm to iron–sulfur clusters relevant for elucidating the mode of action of metalloenzymes.


2020 ◽  
Vol 27 (3) ◽  
pp. 453-471
Author(s):  
Dylan Harries ◽  
Terence J. O'Kane

Abstract. An initial dimension reduction forms an integral part of many analyses in climate science. Different methods yield low-dimensional representations that are based on differing aspects of the data. Depending on the features of the data that are relevant for a given study, certain methods may be more suitable than others, for instance yielding bases that can be more easily identified with physically meaningful modes. To illustrate the distinction between particular methods and identify circumstances in which a given method might be preferred, in this paper we present a set of case studies comparing the results obtained using the traditional approaches of empirical orthogonal function analysis and k-means clustering with the more recently introduced methods such as archetypal analysis and convex coding. For data such as global sea surface temperature anomalies, in which there is a clear, dominant mode of variability, all of the methods considered yield rather similar bases with which to represent the data while differing in reconstruction accuracy for a given basis size. However, in the absence of such a clear scale separation, as in the case of daily geopotential height anomalies, the extracted bases differ much more significantly between the methods. We highlight the importance in such cases of carefully considering the relevant features of interest and of choosing the method that best targets precisely those features so as to obtain more easily interpretable results.


2020 ◽  
Author(s):  
Dylan Harries ◽  
Terence J. O'Kane

Abstract. An initial dimension reduction forms an integral part of many analyses in climate science. Different methods yield low-dimensional representations that are based on differing aspects of the data. Depending on the features of the data that are relevant for a given study, certain methods may be more suitable than others, for instance yielding bases that can be more easily identified with physically meaningful modes. To illustrate the distinction between particular methods and identify circumstances in which a given method might be preferred, in this paper we present a set of case studies comparing the results obtained using the traditional approaches of EOF analysis and k-means clustering with the more recently introduced methods such as archetypal analysis and convex coding. For data such as global sea surface temperature anomalies, in which there is a clear, dominant mode of variability, all of the methods considered yield rather similar bases with which to represent the data, while differing in reconstruction accuracy for a given basis size. However, in the absence of such a clear scale separation, as in the case of daily geopotential height anomalies, the extracted bases differ much more significantly between the methods. We highlight the importance in such cases of carefully considering the relevant features of interest, and of choosing the method that best targets precisely those features so as to obtain more easily interpretable results.


2020 ◽  
Vol 50 (3) ◽  
pp. 537-558 ◽  
Author(s):  
Joseph Skitka ◽  
J. B. Marston ◽  
Baylor Fox-Kemper

AbstractThe combined effectiveness of model reduction and the quasilinear approximation for the reproduction of the low-order statistics of oceanic surface boundary layer turbulence is investigated. Idealized horizontally homogeneous problems of surface-forced thermal convection and Langmuir turbulence are studied in detail. Model reduction is achieved with a Galerkin projection of the governing equations onto a subset of modes determined by proper orthogonal decomposition (POD). When applied to boundary layers that are horizontally homogeneous, POD and a horizontally averaged quasilinear approximation both assume flow features that are horizontally wavelike, making the pairing very efficient. For less than 0.2% of the modes retained, the reduced quasilinear model is able to reproduce vertical profiles of horizontal mean fields as well as certain energetically important second-order turbulent transport statistics and energies to within 30% error. Reduced-basis quasilinear statistics must approach the full-basis statistics as the basis size approaches completion; however, some quasilinear statistics resemble those found in the fully nonlinear simulations at smaller basis truncations. Thus, model reduction could possibly improve upon the accuracy of quasilinear dynamics.


2019 ◽  
Vol 5 (1) ◽  
Author(s):  
Ryan Babbush ◽  
Dominic W. Berry ◽  
Jarrod R. McClean ◽  
Hartmut Neven

Abstract We present a quantum algorithm for simulating quantum chemistry with gate complexity $$\tilde {\cal{O}}(N^{1/3}\eta ^{8/3})$$ O ̃ ( N 1 ∕ 3 η 8 ∕ 3 ) where η is the number of electrons and N is the number of plane wave orbitals. In comparison, the most efficient prior algorithms for simulating electronic structure using plane waves (which are at least as efficient as algorithms using any other basis) have complexity $$\tilde {\cal{O}}(N^{8/3}{\mathrm{/}}\eta ^{2/3})$$ O ̃ ( N 8 ∕ 3 ∕ η 2 ∕ 3 ) . We achieve our scaling in first quantization by performing simulation in the rotating frame of the kinetic operator using interaction picture techniques. Our algorithm is far more efficient than all prior approaches when N ≫ η, as is needed to suppress discretization error when representing molecules in the plane wave basis, or when simulating without the Born-Oppenheimer approximation.


Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 253 ◽  
Author(s):  
Aditya Kamath ◽  
Sergei Manzhos

We explore the use of inverse multiquadratic (IMQ) functions as basis functions when solving the vibrational Schrödinger equation with the rectangular collocation method. The quality of the vibrational spectrum of formaldehyde (in six dimensions) is compared to that obtained using Gaussian basis functions when using different numbers of width-optimized IMQ functions. The effects of the ratio of the number of collocation points to the number of basis functions and of the choice of the IMQ exponent are studied. We show that the IMQ basis can be used with parameters where the IMQ function is not integrable. We find that the quality of the spectrum with IMQ basis functions is somewhat lower that that with a Gaussian basis when the basis size is large, and for a range of IMQ exponents. The IMQ functions are; however, advantageous when a small number of functions is used or with a small number of collocation points (e.g., when using square collocation).


Author(s):  
Aditya Kamath ◽  
Sergei Manzhos

We explore the use of inverse multiquadratic (IMQ) functions as basis functions when solving the vibrational Schrödinger equation with the rectangular collocation method. The quality of the vibrational spectrum of formaldehyde (in six dimensions) is compared to that obtained using Gaussian basis functions when using different numbers of width-optimized IMQ functions. The effects of the ratio of the number of collocation points to the number of basis functions and of the choice of the IMQ exponent are studied. We show that the IMQ basis can be used with parameters where the IMQ function is not integrable. We find that the quality of the spectrum with IMQ basis functions is somewhat lower that that with a Gaussian basis when the basis size is large and for a range of IMQ exponents. The IMQ functions are, however, advantageous when a small number of functions is used or with a small number of collocation points e.g. when using square collocation.


Author(s):  
Г.П. Ганапатхи ◽  
В.Б. Заалишвили ◽  
Д.А. Мельков ◽  
Б.В. Дзеранов ◽  
С.С. Чандрашекаран

Урбанизированные территории, сложенные аллювиальными грунтами, характеризуются уязвимостью к их разжижению даже при землетрясениях средней величины. Разжижение является мерой склонности водонасыщенных отложений к уплотнению во время землетрясения и, таким образом, создает давление поровой воды, достаточное для возможной нестабильности грунта или его разрушения. Здания, построенные на подобных разжижаемых грунтах, весьма уязвимы при колебаниях, обусловленных землетрясением. Город Ченнаи в Индии является одним из самых густонаселенных городов в мире. Застройка, на его большей части, состоит из тесно расположенных высотных зданий. Город находится в пределах умеренной сейсмической зоны и по классификации Бюро Индийского стандарта здесь можно ожидать максимальную величину землетрясения с магнитудой 6,9. Большая часть города, покрытая молодыми аллювиальными грунтами с неглубоким уровнем грунтовых вод, весьма уязвимая при землетрясении, никак не выделяется по внешним признакам. В связи с этим для оценки подверженности грунтов разжижению, в городе проведены исследования на основе изучения геотехнических параметров. Результаты исследования показывают, что более 60% территории городской площади Ченнаи подвержено разжижению. Город Владикавказ в России – один из наиболее плотно населенных городов на Северном Кавказе. Несмотря на отсутствие исторических данных по разжижению грунтов на этой территории, относительно недавно урбанизированной (по крайней мере, в 1810 г.), здесь присутствуют грунты с возможным проявлением явления разжижения при сильных землетрясениях. При этом необходимо учитывать, что непосредственно в южной части города расположен Владикавказский разлом с ожидаемым сейсмическим потенциалом Mmax=7,1. В сотрудничестве с индийскими коллегами метод оценки подверженности грунтов разжижению был адаптирован и применен для территории г. Владикавказа. В то же время в отличие от метода пенетрации (SPT), при исследованиях грунтов Владикавказа использовался более традиционный для России подход, и расчеты были сделаны на основе учета величины скоростей поперечных волн в грунтах. В результате расчетов было установлено, что почти 20% территории города Владикавказа сложено грунтами, подверженных разжижению. Настоящее исследование может заставить градостроительные службы и лиц, принимающих решения, а также аварийно-спасательные службы в их будущей деятельности по планированию развития городских территорий уделять большее внимание подверженности грунтов разжижению. Urban areas lying in the alluvial soil generally pose to threat of liquefaction even for moderate magnitude earthquakes. Liquefaction is the measure of vulnerability of saturated sediment to compact during earthquake shaking and thus generate pore water pressures sufficient to cause possible ground instability or failure. The buildings which are constructed over the liquefiable soil are more vulnerable during seismic shaking for a potential earthquake. The Chennai city of India is one of the most densely populated cities in the world, which consist of densely constructed high rise buildings in many parts. The city is under moderate seismic zone as classified by Bureau of Indian Standard where one can expected maximum magnitude of 6,9. The major part of the city covered by the Recent Alluvial soil with shallow water table, which is more vulnerable during earthquake shaking and quiet enough to trigger liquefaction. In this regard a study carried out to understand the liquefaction susceptibility of soil in the city using geotechnical parameters. Also the study reveals spatially 60% of the area is prone to liquefaction. Vladikavkaz city of Russia is also one of the most densely populated in the North Caucasus. Despite on the absence of historical data on liquefaction on this territory, there are soil conditions in new regions with a possible liquefaction behavior during strong earthquakes. Especially taking into account of Vladikavkaz seismic fault potential of Mmax=7,1. In cooperation with Indian colleagues liquefaction susceptibility assessment method was adopted and applied for Vladikavkaz city. Seismic refraction survey is wide used in Russia rather than SPT and calculations were made on the basis size of shear velocity Vs. As a result 20% of the territory of Vladikavkaz city is liquefiable. The present study can be an eye opening for urban planners and decision makers and emergency responders for future developmental planning activity within the city


Author(s):  
Joël Cathébras ◽  
Alexandre Carbon ◽  
Peter Milder ◽  
Renaud Sirdey ◽  
Nicolas Ventroux

This paper presents a hardware implementation of a Residue Polynomial Multiplier (RPM), designed to accelerate the full Residue Number System (RNS) variant of the Fan-Vercauteren scheme proposed by Bajard et al. [BEHZ16]. Our design speeds up polynomial multiplication via a Negative Wrapped Convolution (NWC) which locally computes the required RNS channel dependent twiddle factors. Compared to related works, this design is more versatile regarding the addressable parameter sets for the BFV scheme. This is mainly brought by our proposed twiddle factor generator that makes the design BRAM utilization independent of the RNS basis size, with a negligible communication bandwidth usage for non-payload data. Furthermore, the generalization of a DFT hardware generator is explored in order to generate RNS friendly NTT architectures. This approach helps us to validate our RPM design over parameter sets from the work of Halevi et al. [HPS18]. For the depth-20 setting, we achieve an estimated speed up for the residue polynomial multiplications greater than 76 during ciphertexts multiplication, and greater than 16 during relinearization. It thus results in a single-threaded Mult&Relin ciphertext operation in 109.4 ms (×3.19 faster than [HPS18]) with RPM counting for less than 15% of the new computation time. Our RPM design scales up with reasonable use of hardware resources and realistic bandwidth requirements. It can also be exploited for other RNS based implementations of RLWE cryptosystems.


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