19 research outputs found
Normalizer Circuits and Quantum Computation
(Abridged abstract.) In this thesis we introduce new models of quantum
computation to study the emergence of quantum speed-up in quantum computer
algorithms.
Our first contribution is a formalism of restricted quantum operations, named
normalizer circuit formalism, based on algebraic extensions of the qubit
Clifford gates (CNOT, Hadamard and -phase gates): a normalizer circuit
consists of quantum Fourier transforms (QFTs), automorphism gates and quadratic
phase gates associated to a set , which is either an abelian group or
abelian hypergroup. Though Clifford circuits are efficiently classically
simulable, we show that normalizer circuit models encompass Shor's celebrated
factoring algorithm and the quantum algorithms for abelian Hidden Subgroup
Problems. We develop classical-simulation techniques to characterize under
which scenarios normalizer circuits provide quantum speed-ups. Finally, we
devise new quantum algorithms for finding hidden hyperstructures. The results
offer new insights into the source of quantum speed-ups for several algebraic
problems.
Our second contribution is an algebraic (group- and hypergroup-theoretic)
framework for describing quantum many-body states and classically simulating
quantum circuits. Our framework extends Gottesman's Pauli Stabilizer Formalism
(PSF), wherein quantum states are written as joint eigenspaces of stabilizer
groups of commuting Pauli operators: while the PSF is valid for qubit/qudit
systems, our formalism can be applied to discrete- and continuous-variable
systems, hybrid settings, and anyonic systems. These results enlarge the known
families of quantum processes that can be efficiently classically simulated.
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 codes.Comment: PhD thesis, Technical University of Munich (2016). Please cite
original papers if possible. Appendix E contains unpublished work on Gaussian
unitaries. If you spot typos/omissions please email me at JLastNames at
posteo dot net. Source: http://bit.ly/2gMdHn3. Related video talk:
https://www.perimeterinstitute.ca/videos/toy-theory-quantum-speed-ups-based-stabilizer-formalism
Posted on my birthda
Helix-Hopes on Finite Hyperfields
Hyperstructure theory can overcome restrictions which ordinary algebraic structures have. A hyperproduct on non-square ordinary matrices can be defined by using the so called helix-hyperoperations. We study the helix-hyperstructures on the representations using ordinary fields. The related theory can be faced by defining the hyperproduct on the set of non square matrices. The main tools of the Hyperstructure Theory are the fundamental relations which connect the largest class of hyperstructures, the Hv-structures, with the corresponding classical ones. We focus on finite dimensional helix-hyperstructures and on small Hv-fields, as well.
On power structures
In general, power structure of a structure A (with
universe A) is an appropriate structure defined on the power set
P(A). There are lot of papers concerning this topic in which power
structures appear explicitly or implicitly. The aim of this paper
is to give an overview of the results that are interesting from the
universal-algebraic point of view
Collected Papers (on Neutrosophic Theory and Its Applications in Algebra), Volume IX
This ninth volume of Collected Papers includes 87 papers comprising 982 pages on Neutrosophic Theory and its applications in Algebra, written between 2014-2022 by the author alone or in collaboration with the following 81 co-authors (alphabetically ordered) from 19 countries: E.O. Adeleke, A.A.A. Agboola, Ahmed B. Al-Nafee, Ahmed Mostafa Khalil, Akbar Rezaei, S.A. Akinleye, Ali Hassan, Mumtaz Ali, Rajab Ali Borzooei , Assia Bakali, Cenap Özel, Victor Christianto, Chunxin Bo, Rakhal Das, Bijan Davvaz, R. Dhavaseelan, B. Elavarasan, Fahad Alsharari, T. Gharibah, Hina Gulzar, Hashem Bordbar, Le Hoang Son, Emmanuel Ilojide, Tèmítópé Gbóláhàn Jaíyéolá, M. Karthika, Ilanthenral Kandasamy, W.B. Vasantha Kandasamy, Huma Khan, Madad Khan, Mohsin Khan, Hee Sik Kim, Seon Jeong Kim, Valeri Kromov, R. M. Latif, Madeleine Al-Tahan, Mehmat Ali Ozturk, Minghao Hu, S. Mirvakili, Mohammad Abobala, Mohammad Hamidi, Mohammed Abdel-Sattar, Mohammed A. Al Shumrani, Mohamed Talea, Muhammad Akram, Muhammad Aslam, Muhammad Aslam Malik, Muhammad Gulistan, Muhammad Shabir, G. Muhiuddin, Memudu Olaposi Olatinwo, Osman Anis, Choonkil Park, M. Parimala, Ping Li, K. Porselvi, D. Preethi, S. Rajareega, N. Rajesh, Udhayakumar Ramalingam, Riad K. Al-Hamido, Yaser Saber, Arsham Borumand Saeid, Saeid Jafari, Said Broumi, A.A. Salama, Ganeshsree Selvachandran, Songtao Shao, Seok-Zun Song, Tahsin Oner, M. Mohseni Takallo, Binod Chandra Tripathy, Tugce Katican, J. Vimala, Xiaohong Zhang, Xiaoyan Mao, Xiaoying Wu, Xingliang Liang, Xin Zhou, Yingcang Ma, Young Bae Jun, Juanjuan Zhang
Algebraic geometry over semi-structures and hyper-structures of characteristic one
In this thesis, we study algebraic geometry in characteristic one from the perspective of semirings
and hyperrings. The thesis largely consists of three parts:
(1) We develop the basic notions and several methods of algebraic geometry over semirings. We first construct a semi-scheme by directly generalizing the classical construction of a scheme, and prove that any semiring can be canonically realized as a semiring of global functions on an affine semischeme. We then develop Cech cohomology theory for semi-schemes, and show that the classical
isomorphism is still valid for a semi-scheme. Finally, we introduce the notion of a valuation on a semiring, and prove that an analogue of an abstract curve by using the (suitably defined) function field Qmax(T) is homeomorphic to the projective line over the field with one element. (2) We develop algebraic geometry over hyperrings. The first motivation for this study arises from the following problem posed in [9]: if one follows the classical construction to define the hyper-scheme (X = SpecR,O_X), where R is a hyperring, then a canonical isomorphism R ≃ O_X(X) does not hold in general. By investigating algebraic properties of hyperrings (which include a construction of a quotient hyperring and Hilbert Nullstellensatz), we give a partial answer for their problem as follows: when R does not have a (multiplicative) zero-divisor, the canonical isomorphism R ≃ O_X(X) holds for a hyper-scheme (X = SpecR,OX). In other words, R can be realized as a hyperring of global functions on an affine hyper-scheme.
We also give a (partial) affirmative answer to the following speculation posed by Connes and Consani
in [7]: let A = k[T] or k[T, 1/T ], where k = Q or Fp. When k = Fp, the topological space SpecA
is a hypergroup with a canonical hyper-operation ∗ induced from a coproduct of A. The similar
statement holds with k = Q and SpecA\{δ}, where δ is the generic point (cf. [7, Theorems 7.1
and 7.13]). Connes and Consani expected that the similar result would be true for Chevalley group schemes. We prove that when X = SpecA is an affine algebraic group scheme over arbitrary field, then, together with a canonical hyper-operation ∗ on X introduced in [7], (X, ∗) becomes a slightly general (in a precise sense) object than a hypergroup. (3) We give a (partial) converse of S.Henry’s symmetrization procedure which produces a hypergroup from a semigroup in a canonical way (cf. [21]). Furthermore, via the symmetrization process, we connect the notions of (1) and (2), and prove that such a link is closely related with the notion of real prime ideals
Extensions of Positive Definite Functions: Applications and Their Harmonic Analysis
We study two classes of extension problems, and their interconnections: (i)
Extension of positive definite (p.d.) continuous functions defined on subsets
in locally compact groups ; (ii) In case of Lie groups, representations of
the associated Lie algebras by unbounded skew-Hermitian
operators acting in a reproducing kernel Hilbert space (RKHS)
.
Why extensions? In science, experimentalists frequently gather spectral data
in cases when the observed data is limited, for example limited by the
precision of instruments; or on account of a variety of other limiting external
factors. Given this fact of life, it is both an art and a science to still
produce solid conclusions from restricted or limited data. In a general sense,
our monograph deals with the mathematics of extending some such given partial
data-sets obtained from experiments. More specifically, we are concerned with
the problems of extending available partial information, obtained, for example,
from sampling. In our case, the limited information is a restriction, and the
extension in turn is the full positive definite function (in a dual variable);
so an extension if available will be an everywhere defined generating function
for the exact probability distribution which reflects the data; if it were
fully available. Such extensions of local information (in the form of positive
definite functions) will in turn furnish us with spectral information. In this
form, the problem becomes an operator extension problem, referring to operators
in a suitable reproducing kernel Hilbert spaces (RKHS). In our presentation we
have stressed hands-on-examples. Extensions are almost never unique, and so we
deal with both the question of existence, and if there are extensions, how they
relate back to the initial completion problem.Comment: 235 pages, 42 figures, 7 tables. arXiv admin note: substantial text
overlap with arXiv:1401.478
Towers of recollement and bases for diagram algebras: planar diagrams and a little beyond
The recollement approach to the representation theory of sequences of
algebras is extended to pass basis information directly through the
globalisation functor. The method is hence adapted to treat sequences that are
not necessarily towers by inclusion, such as symplectic blob algebras (diagram
algebra quotients of the type-\hati{C} Hecke algebras).
By carefully reviewing the diagram algebra construction, we find a new set of
functors interrelating module categories of ordinary blob algebras (diagram
algebra quotients of the type- Hecke algebras) at {\em different} values
of the algebra parameters. We show that these functors generalise to determine
the structure of symplectic blob algebras, and hence of certain two-boundary
Temperley-Lieb algebras arising in Statistical Mechanics.
We identify the diagram basis with a cellular basis for each symplectic blob
algebra, and prove that these algebras are quasihereditary over a field for
almost all parameter choices, and generically semisimple. (That is, we give
bases for all cell and standard modules.)Comment: 61 page
Neutrosophic SuperHyperAlgebra and New Types of Topologies
In general, a system S (that may be a company, association, institution, society, country, etc.) is formed by sub-systems Si { or P(S), the powerset of S }, and each sub-system Si is formed by sub-sub-systems Sij { or P(P(S)) = P2(S) } and so on. That’s why the n-th PowerSet of a Set S { defined recursively and denoted by Pn(S) = P(Pn-1(S) } was introduced, to better describes the organization of people, beings, objects etc. in our real world. The n-th PowerSet was used in defining the SuperHyperOperation, SuperHyperAxiom, and their corresponding Neutrosophic SuperHyperOperation, Neutrosophic SuperHyperAxiom in order to build the SuperHyperAlgebra and Neutrosophic SuperHyperAlgebra. In general, in any field of knowledge, one in fact encounters SuperHyperStructures. Also, six new types of topologies have been introduced in the last years (2019-2022), such as: Refined Neutrosophic Topology, Refined Neutrosophic Crisp Topology, NeutroTopology, AntiTopology, SuperHyperTopology, and Neutrosophic SuperHyperTopology