Universality in Quantum Computation
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Real Composition Algebras by Steven Clanton
Real Composition Algebras by Steven Clanton A Thesis Submitted to the Faculty of The Wilkes Honors College in Partial Fulfillment of the Requirements for the Degree of Bachelor of Arts in Liberal Arts and Sciences with a Concentration in Mathematics Wilkes Honors College of Florida Atlantic University Jupiter, FL May 2009 Real Composition Algebras by Steven Clanton This thesis was prepared under the direction of the candidates thesis advisor, Dr. Ryan Karr, and has been approved by the members of his supervisory committee. It was sub- mitted to the faculty of The Honors College and was accepted in partial fulfillment of the requirements for the degree of Bachelor of Arts in Liberal Arts and Sciences. SUPERVISORY COMMITTEE: Dr. Ryan Karr Dr. Eugene Belogay Dean, Wilkes Honors College Date ii Abstract Author: Steve Clanton Title: Real Composition Algebras Institution: Wilkes Honors College of Florida Atlantic University Thesis Advisor: Dr. Ryan Karr Degree: Bachelor of Arts in Liberal Arts and Sciences Concentration: Mathematics Year: 2009 According to Koecher and Remmert [Ebb91, p. 267], Gauss introduced the \com- position of quadratic forms" to study the representability of natural numbers in binary (rank 2) quadratic forms. In particular, Gauss proved that any positive defi- nite binary quadratic form can be linearly transformed into the two-squares formula 2 2 2 2 2 2 w1 + w2 = (u1 + u2)(v1 + v2). This shows not only the existence of an algebra for every form but also an isomorphism to the complex numbers for each. Hurwitz gen- eralized the \theory of composition" to arbitrary dimensions and showed that exactly four such systems exist: the real numbers, the complex numbers, the quaternions, and octonions [CS03, p. -
The Classification and the Conjugacy Classesof the Finite Subgroups of The
Algebraic & Geometric Topology 8 (2008) 757–785 757 The classification and the conjugacy classes of the finite subgroups of the sphere braid groups DACIBERG LGONÇALVES JOHN GUASCHI Let n 3. We classify the finite groups which are realised as subgroups of the sphere 2 braid group Bn.S /. Such groups must be of cohomological period 2 or 4. Depend- ing on the value of n, we show that the following are the maximal finite subgroups of 2 Bn.S /: Z2.n 1/ ; the dicyclic groups of order 4n and 4.n 2/; the binary tetrahedral group T ; the binary octahedral group O ; and the binary icosahedral group I . We give geometric as well as some explicit algebraic constructions of these groups in 2 Bn.S / and determine the number of conjugacy classes of such finite subgroups. We 2 also reprove Murasugi’s classification of the torsion elements of Bn.S / and explain 2 how the finite subgroups of Bn.S / are related to this classification, as well as to the 2 lower central and derived series of Bn.S /. 20F36; 20F50, 20E45, 57M99 1 Introduction The braid groups Bn of the plane were introduced by E Artin in 1925[2;3]. Braid groups of surfaces were studied by Zariski[41]. They were later generalised by Fox to braid groups of arbitrary topological spaces via the following definition[16]. Let M be a compact, connected surface, and let n N . We denote the set of all ordered 2 n–tuples of distinct points of M , known as the n–th configuration space of M , by: Fn.M / .p1;:::; pn/ pi M and pi pj if i j : D f j 2 ¤ ¤ g Configuration spaces play an important roleˆ in several branches of mathematics and have been extensively studied; see Cohen and Gitler[9] and Fadell and Husseini[14], for example. -
INTEGRAL CAYLEY GRAPHS and GROUPS 3 of G on W
INTEGRAL CAYLEY GRAPHS AND GROUPS AZHVAN AHMADY, JASON P. BELL, AND BOJAN MOHAR Abstract. We solve two open problems regarding the classification of certain classes of Cayley graphs with integer eigenvalues. We first classify all finite groups that have a “non-trivial” Cayley graph with integer eigenvalues, thus solving a problem proposed by Abdollahi and Jazaeri. The notion of Cayley integral groups was introduced by Klotz and Sander. These are groups for which every Cayley graph has only integer eigenvalues. In the second part of the paper, all Cayley integral groups are determined. 1. Introduction A graph X is said to be integral if all eigenvalues of the adjacency matrix of X are integers. This property was first defined by Harary and Schwenk [9] who suggested the problem of classifying integral graphs. This problem ignited a signifi- cant investigation among algebraic graph theorists, trying to construct and classify integral graphs. Although this problem is easy to state, it turns out to be extremely hard. It has been attacked by many mathematicians during the last forty years and it is still wide open. Since the general problem of classifying integral graphs seems too difficult, graph theorists started to investigate special classes of graphs, including trees, graphs of bounded degree, regular graphs and Cayley graphs. What proves so interesting about this problem is that no one can yet identify what the integral trees are or which 5-regular graphs are integral. The notion of CIS groups, that is, groups admitting no integral Cayley graphs besides complete multipartite graphs, was introduced by Abdollahi and Jazaeri [1], who classified all abelian CIS groups. -
Generalized Quaternions
GENERALIZED QUATERNIONS KEITH CONRAD 1. introduction The quaternion group Q8 is one of the two non-abelian groups of size 8 (up to isomor- phism). The other one, D4, can be constructed as a semi-direct product: ∼ ∼ × ∼ D4 = Aff(Z=(4)) = Z=(4) o (Z=(4)) = Z=(4) o Z=(2); where the elements of Z=(2) act on Z=(4) as the identity and negation. While Q8 is not a semi-direct product, it can be constructed as the quotient group of a semi-direct product. We will see how this is done in Section2 and then jazz up the construction in Section3 to make an infinite family of similar groups with Q8 as the simplest member. In Section4 we will compare this family with the dihedral groups and see how it fits into a bigger picture. 2. The quaternion group from a semi-direct product The group Q8 is built out of its subgroups hii and hji with the overlapping condition i2 = j2 = −1 and the conjugacy relation jij−1 = −i = i−1. More generally, for odd a we have jaij−a = −i = i−1, while for even a we have jaij−a = i. We can combine these into the single formula a (2.1) jaij−a = i(−1) for all a 2 Z. These relations suggest the following way to construct the group Q8. Theorem 2.1. Let H = Z=(4) o Z=(4), where (a; b)(c; d) = (a + (−1)bc; b + d); ∼ The element (2; 2) in H has order 2, lies in the center, and H=h(2; 2)i = Q8. -
Module 2 : Fundamentals of Vector Spaces Section 2 : Vector Spaces
Module 2 : Fundamentals of Vector Spaces Section 2 : Vector Spaces 2. Vector Spaces The concept of a vector space will now be formally introduced. This requires the concept of closure and field. Definition 1 (Closure) A set is said to be closed under an operation when any two elements of the set subject to the operation yields a third element belonging to the same set. Example 2 The set of integers is closed under addition, multiplication and subtraction. However, this set is not closed under division. Example 3 The set of real numbers ( ) and the set of complex numbers (C) are closed under addition, subtraction, multiplication and division. Definition 4 (Field) A field is a set of elements closed under addition, subtraction, multiplication and division. Example 5 The set of real numbers ( ) and the set of complex numbers ( ) are scalar fields. However, the set of integers is not a field. A vector space is a set of elements, which is closed under addition and scalar multiplication. Thus, associated with every vector space is a set of scalars (also called as scalar field or coefficient field) used to define scalar multiplication on the space. In functional analysis, the scalars will be always taken to be the set of real numbers ( ) or complex numbers ( ). Definition 6 (Vector Space): A vector space is a set of elements called vectors and scalar field together with two operations. The first operation is called addition which associates with any two vectors a vector , the sum of and . The second operation is called scalar multiplication, which associates with any vector and any scalar a vector (a scalar multiple of by Thus, when is a linear vector space, given any vectors and any scalars the element . -
12.6 Further Topics on Simple Groups 387 12.6 Further Topics on Simple Groups
12.6 Further Topics on Simple groups 387 12.6 Further Topics on Simple Groups This Web Section has three parts (a), (b) and (c). Part (a) gives a brief descriptions of the 56 (isomorphism classes of) simple groups of order less than 106, part (b) provides a second proof of the simplicity of the linear groups Ln(q), and part (c) discusses an ingenious method for constructing a version of the Steiner system S(5, 6, 12) from which several versions of S(4, 5, 11), the system for M11, can be computed. 12.6(a) Simple Groups of Order less than 106 The table below and the notes on the following five pages lists the basic facts concerning the non-Abelian simple groups of order less than 106. Further details are given in the Atlas (1985), note that some of the most interesting and important groups, for example the Mathieu group M24, have orders in excess of 108 and in many cases considerably more. Simple Order Prime Schur Outer Min Simple Order Prime Schur Outer Min group factor multi. auto. simple or group factor multi. auto. simple or count group group N-group count group group N-group ? A5 60 4 C2 C2 m-s L2(73) 194472 7 C2 C2 m-s ? 2 A6 360 6 C6 C2 N-g L2(79) 246480 8 C2 C2 N-g A7 2520 7 C6 C2 N-g L2(64) 262080 11 hei C6 N-g ? A8 20160 10 C2 C2 - L2(81) 265680 10 C2 C2 × C4 N-g A9 181440 12 C2 C2 - L2(83) 285852 6 C2 C2 m-s ? L2(4) 60 4 C2 C2 m-s L2(89) 352440 8 C2 C2 N-g ? L2(5) 60 4 C2 C2 m-s L2(97) 456288 9 C2 C2 m-s ? L2(7) 168 5 C2 C2 m-s L2(101) 515100 7 C2 C2 N-g ? 2 L2(9) 360 6 C6 C2 N-g L2(103) 546312 7 C2 C2 m-s L2(8) 504 6 C2 C3 m-s -
Euclidean Space - Wikipedia, the Free Encyclopedia Page 1 of 5
Euclidean space - Wikipedia, the free encyclopedia Page 1 of 5 Euclidean space From Wikipedia, the free encyclopedia In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions. The term “Euclidean” distinguishes these spaces from the curved spaces of non-Euclidean geometry and Einstein's general theory of relativity, and is named for the Greek mathematician Euclid of Alexandria. Classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. In modern mathematics, it is more common to define Euclidean space using Cartesian coordinates and the ideas of analytic geometry. This approach brings the tools of algebra and calculus to bear on questions of geometry, and Every point in three-dimensional has the advantage that it generalizes easily to Euclidean Euclidean space is determined by three spaces of more than three dimensions. coordinates. From the modern viewpoint, there is essentially only one Euclidean space of each dimension. In dimension one this is the real line; in dimension two it is the Cartesian plane; and in higher dimensions it is the real coordinate space with three or more real number coordinates. Thus a point in Euclidean space is a tuple of real numbers, and distances are defined using the Euclidean distance formula. Mathematicians often denote the n-dimensional Euclidean space by , or sometimes if they wish to emphasize its Euclidean nature. Euclidean spaces have finite dimension. Contents 1 Intuitive overview 2 Real coordinate space 3 Euclidean structure 4 Topology of Euclidean space 5 Generalizations 6 See also 7 References Intuitive overview One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance and angle. -
Math 20480 – Example Set 03A Determinant of a Square Matrices
Math 20480 { Example Set 03A Determinant of A Square Matrices. Case: 2 × 2 Matrices. a b The determinant of A = is c d a b det(A) = jAj = = c d For matrices with size larger than 2, we need to use cofactor expansion to obtain its value. Case: 3 × 3 Matrices. 0 1 a11 a12 a13 a11 a12 a13 The determinant of A = @a21 a22 a23A is det(A) = jAj = a21 a22 a23 a31 a32 a33 a31 a32 a33 (Cofactor expansion along the 1st row) = a11 − a12 + a13 (Cofactor expansion along the 2nd row) = (Cofactor expansion along the 1st column) = (Cofactor expansion along the 2nd column) = Example 1: 1 1 2 ? 1 −2 −1 = 1 −1 1 1 Case: 4 × 4 Matrices. a11 a12 a13 a14 a21 a22 a23 a24 a31 a32 a33 a34 a41 a42 a43 a44 (Cofactor expansion along the 1st row) = (Cofactor expansion along the 2nd column) = Example 2: 1 −2 −1 3 −1 2 0 −1 ? = 0 1 −2 2 3 −1 2 −3 2 Theorem 1: Let A be a square n × n matrix. Then the A has an inverse if and only if det(A) 6= 0. We call a matrix A with inverse a matrix. Theorem 2: Let A be a square n × n matrix and ~b be any n × 1 matrix. Then the equation A~x = ~b has a solution for ~x if and only if . Proof: Remark: If A is non-singular then the only solution of A~x = 0 is ~x = . 3 3. Without solving explicitly, determine if the following systems of equations have a unique solution. -
Supplement. Direct Products and Semidirect Products
Supplement: Direct Products and Semidirect Products 1 Supplement. Direct Products and Semidirect Products Note. In Section I.8 of Hungerford, we defined direct products and weak direct n products. Recall that when dealing with a finite collection of groups {Gi}i=1 then the direct product and weak direct product coincide (Hungerford, page 60). In this supplement we give results concerning recognizing when a group is a direct product of smaller groups. We also define the semidirect product and illustrate its use in classifying groups of small order. The content of this supplement is based on Sections 5.4 and 5.5 of Davis S. Dummitt and Richard M. Foote’s Abstract Algebra, 3rd Edition, John Wiley and Sons (2004). Note. Finitely generated abelian groups are classified in the Fundamental Theo- rem of Finitely Generated Abelian Groups (Theorem II.2.1). So when addressing direct products, we are mostly concerned with nonabelian groups. Notice that the following definition is “dull” if applied to an abelian group. Definition. Let G be a group, let x, y ∈ G, and let A, B be nonempty subsets of G. (1) Define [x, y]= x−1y−1xy. This is the commutator of x and y. (2) Define [A, B]= h[a,b] | a ∈ A,b ∈ Bi, the group generated by the commutators of elements from A and B where the binary operation is the same as that of group G. (3) Define G0 = {[x, y] | x, y ∈ Gi, the subgroup of G generated by the commuta- tors of elements from G under the same binary operation of G. -
Automatic Variational Inference in Stan
Automatic Variational Inference in Stan Alp Kucukelbir Data Science Institute Department of Computer Science Columbia University [email protected] Rajesh Ranganath Department of Computer Science Princeton University [email protected] Andrew Gelman Data Science Institute Depts. of Political Science, Statistics Columbia University [email protected] David M. Blei Data Science Institute Depts. of Computer Science, Statistics Columbia University [email protected] June 15, 2015 Abstract Variational inference is a scalable technique for approximate Bayesian inference. Deriving variational inference algorithms requires tedious model-specific calculations; this makes it difficult to automate. We propose an automatic variational inference al- gorithm, automatic differentiation variational inference (advi). The user only provides a Bayesian model and a dataset; nothing else. We make no conjugacy assumptions and arXiv:1506.03431v2 [stat.ML] 12 Jun 2015 support a broad class of models. The algorithm automatically determines an appropri- ate variational family and optimizes the variational objective. We implement advi in Stan (code available now), a probabilistic programming framework. We compare advi to mcmc sampling across hierarchical generalized linear models, nonconjugate matrix factorization, and a mixture model. We train the mixture model on a quarter million images. With advi we can use variational inference on any model we write in Stan. 1 Introduction Bayesian inference is a powerful framework for analyzing data. We design -
Universality of Single-Qudit Gates
Ann. Henri Poincar´e Online First c 2017 The Author(s). This article is an open access publication Annales Henri Poincar´e DOI 10.1007/s00023-017-0604-z Universality of Single-Qudit Gates Adam Sawicki and Katarzyna Karnas Abstract. We consider the problem of deciding if a set of quantum one- qudit gates S = {g1,...,gn}⊂G is universal, i.e. if <S> is dense in G, where G is either the special unitary or the special orthogonal group. To every gate g in S we assign the orthogonal matrix Adg that is image of g under the adjoint representation Ad : G → SO(g)andg is the Lie algebra of G. The necessary condition for the universality of S is that the only matrices that commute with all Adgi ’s are proportional to the identity. If in addition there is an element in <S> whose Hilbert–Schmidt distance √1 S from the centre of G belongs to ]0, 2 [, then is universal. Using these we provide a simple algorithm that allows deciding the universality of any set of d-dimensional gates in a finite number of steps and formulate a general classification theorem. 1. Introduction Quantum computer is a device that operates on a finite-dimensional quantum system H = H1 ⊗···⊗Hn consisting of n qudits [4,19,28] that are described by d d-dimensional Hilbert spaces, Hi C [29]. When d = 2 qudits are typically called qubits. The ability to effectively manufacture optical gates operating on many modes, using, for example, optical networks that couple modes of light [9,33,34], is a natural motivation to consider not only qubits but also higher- dimensional systems in the quantum computation setting (see also [30,31] for the case of fermionic linear optics and quantum metrology). -
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JWST582-IND JWST582-Ray Printer: Yet to Come September 7, 2015 10:27 Trim: 254mm × 178mm Index Adjoint Map and Left and Right Translations, 839–842, 845, 850–854, 857–858, 126 861–862, 880, 885, 906, 913–914, Admissible Virtual Space, 91, 125, 524, 925, 934, 950–953 560–561, 564, 722, 788, 790, 793 Scaling, 37, 137, 161, 164, 166, 249 affine combinations, 166 Translation, 33–34, 42–43, 75, 77, 92, 127, Affine Space, 33–34, 41, 43, 136–137, 139, 136–137, 155, 164, 166–167, 280, 155, 248, 280, 352–353, 355–356 334, 355–356, 373, 397, 421, 533, Affine Transformation 565, 793 Identity, 21–22, 39, 47, 60–61, 63–64, Algorithm For Newton Method of Iteration, 67–68, 70, 81, 90, 98–100, 104–106, 441 116, 125–126, 137, 255, 259, 268, Algorithm: Numerical Iteration, 445 331, 347, 362–363, 367, 372, 375, Allowable Parametric Representation, 37, 377, 380, 398, 538, 550, 559, 562, 140, 249 564–565, 568, 575–576, 581, 599, An Important Lemma, 89, 605, 839 609, 616, 622, 628, 648, 655, 663, Angle between Vectors, 259 668, 670, 730, 733, 788, 790, Angular Momentum, 523–524, 537–538, 543, 792–793, 797–798, 805–806, 547–549, 553–554, 556, 558, 575, 585, 811, 825, 834, 852, 862, 871–872, 591–592, 594, 608, 622, 640, 647–648, 916 722, 748, 750, 755–756, 759–760, 762, Rotation, 1, 6, 9–15, 17, 22, 27, 33, 36, 39, 769–770, 776, 778, 780–783, 787, 805, 44, 64, 74, 90–94, 97–101, 104–108, 815, 822, 828, 842 110, 112, 114–116, 118–121, Angular Velocity Tensor, 125, 127, 526, 542, 123–127, 131, 137, 165–166, 325, 550, 565, 571, 723, 794, 800 334, 336–337, 353, 356,