In computational complexity theory, bounded-error quantum polynomial time (BQP) is the class of decision problems solvable by a quantum computer in polynomial time, with an error probability of at most 1/3 for all instances. It is the quantum analogue to the complexity class BPP.
Bra–ket notation, also called Dirac notation, is a notation for linear algebra and linear operators on complex vector spaces together with their dual space both in the finite-dimensional and infinite-dimensional case. It is specifically designed to ease the types of calculations that frequently come up in quantum mechanics. Its use in quantum mechanics is quite widespread.
In mathematics, a geometric algebra is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the geometric product. Multiplication of vectors results in higher-dimensional objects called multivectors. Compared to other formalisms for manipulating geometric objects, geometric algebra is noteworthy for supporting vector division and addition of objects of different dimensions.
In mathematics, an inner product space is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often denoted with angle brackets such as in . Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality of vectors. Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or scalar product of Cartesian coordinates. Inner product spaces of infinite dimension are widely used in functional analysis. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898.
Linear algebra is the branch of mathematics concerning linear equations such as:
The Riesz representation theorem, sometimes called the Riesz–Fréchet representation theorem after Frigyes Riesz and Maurice René Fréchet, establishes an important connection between a Hilbert space and its continuous dual space. If the underlying field is the real numbers, the two are isometrically isomorphic; if the underlying field is the complex numbers, the two are isometrically anti-isomorphic. The (anti-) isomorphism is a particular natural isomorphism.
The Cauchy–Schwarz inequality is an upper bound on the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is considered one of the most important and widely used inequalities in mathematics.
In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process or Gram-Schmidt algorithm is a way of finding a set of two or more vectors that are perpendicular to each other.
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space with finite dimension is a basis for whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, the standard basis for a Euclidean space is an orthonormal basis, where the relevant inner product is the dot product of vectors. The image of the standard basis under a rotation or reflection is also orthonormal, and every orthonormal basis for arises in this fashion.
In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the algebra produces the Hodge dual of the element. This map was introduced by W. V. D. Hodge.
In linear algebra and functional analysis, a projection is a linear transformation from a vector space to itself such that . That is, whenever is applied twice to any vector, it gives the same result as if it were applied once. It leaves its image unchanged. This definition of "projection" formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object.
Multi-task learning (MTL) is a subfield of machine learning in which multiple learning tasks are solved at the same time, while exploiting commonalities and differences across tasks. This can result in improved learning efficiency and prediction accuracy for the task-specific models, when compared to training the models separately. Early versions of MTL were called "hints".
In linear algebra, the Gram matrix of a set of vectors in an inner product space is the Hermitian matrix of inner products, whose entries are given by the inner product . If the vectors are the columns of matrix then the Gram matrix is in the general case that the vector coordinates are complex numbers, which simplifies to for the case that the vector coordinates are real numbers.
In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. If a norm arises from an inner product then the polarization identity can be used to express this inner product entirely in terms of the norm. The polarization identity shows that a norm can arise from at most one inner product; however, there exist norms that do not arise from any inner product.
Semidefinite programming (SDP) is a subfield of mathematical programming concerned with the optimization of a linear objective function over the intersection of the cone of positive semidefinite matrices with an affine space, i.e., a spectrahedron.
Nearest neighbor search (NNS), as a form of proximity search, is the optimization problem of finding the point in a given set that is closest to a given point. Closeness is typically expressed in terms of a dissimilarity function: the less similar the objects, the larger the function values.
In functional analysis, the dual norm is a measure of size for a continuous linear function defined on a normed vector space.
In cryptography, learning with errors (LWE) is a mathematical problem that is widely used to create secure encryption algorithms. It is based on the idea of representing secret information as a set of equations with errors. In other words, LWE is a way to hide the value of a secret by introducing noise to it. In more technical terms, it refers to the computational problem of inferring a linear -ary function over a finite ring from given samples some of which may be erroneous. The LWE problem is conjectured to be hard to solve, and thus to be useful in cryptography.
In discrete mathematics, ideal lattices are a special class of lattices and a generalization of cyclic lattices. Ideal lattices naturally occur in many parts of number theory, but also in other areas. In particular, they have a significant place in cryptography. Micciancio defined a generalization of cyclic lattices as ideal lattices. They can be used in cryptosystems to decrease by a square root the number of parameters necessary to describe a lattice, making them more efficient. Ideal lattices are a new concept, but similar lattice classes have been used for a long time. For example, cyclic lattices, a special case of ideal lattices, are used in NTRUEncrypt and NTRUSign.
This is a glossary for the terminology in a mathematical field of functional analysis.
