Related papers: Geometric algebra, qubits, geometric evolution, an…
This article provides a simple geometric interpretation of the quadratic formula. The geometry helps to demystify the formula's complex appearance and casts it into a much simpler existence, thus potentially benefits early algebra students.
Generalisations of geometry have emerged in various forms in the study of field theory and quantization. This mini-review focuses on the role of higher geometry in three selected physical applications. After motivating and describing some…
This paper presents a new class of adaptive filters, namely Geometric-Algebra Adaptive Filters (GAAFs). They are generated by formulating the underlying minimization problem (a deterministic cost function) from the perspective of Geometric…
There are a wide variety of different vector formalisms currently utilized in engineering and physics. For example, Gibbs' three-vectors, Minkowski four-vectors, complex spinors in quantum mechanics, quaternions used to describe rigid body…
Problems involving geometric data arise in physics, chemistry, robotics, computer vision, and many other fields. Such data can take numerous forms, for instance points, direction vectors, translations, or rotations, but to date there is no…
We review a geometric approach to classification and examination of quantum correlations in composite systems. Since quantum information tasks are usually achieved by manipulating spin and alike systems or, in general, systems with a finite…
Geometric phases arise in a number of physical situations and often lead to systematic shifts in frequencies or phases measured in precision experiments. We describe, by working through some simple examples, a method to calculate geometric…
The article presents a new approach to euclidean plane geometry based on projective geometric algebra (PGA). It is designed for anyone with an interest in plane geometry, or who wishes to familiarize themselves with PGA. After a brief…
The second quantized approach to geometric phases is reviewed. The second quantization generally induces a hidden local (time-dependent) gauge symmetry. This gauge symmetry defines the parallel transport and holonomy, and thus it controls…
The controlled-NOT gate of qubit quantum circuits is shown to be described in terms of a Hopf algebra. Accordingly, any qubit quantum circuit can be expressed as the Hopf algebraic computations and unitary transformations on one qubit.
A tutorial of the Mathematica package CGAlgebra, for conformal geometric algebra calculations is presented. Using rule-based programming, the 5-dimensional conformal geometric algebra is implemented and defined functions simplify the…
This paper concentrates on the homogeneous (conformal) model of Euclidean space (Horosphere) with subspaces that intuitively correspond to Euclidean geometric objects in three dimensions. Mathematical details of the construction and…
Geometric quantum computation is the idea that geometric phases can be used to implement quantum gates, i.e., the basic elements of the Boolean network that forms a quantum computer. Although originally thought to be limited to adiabatic…
Computational complexity is a new quantum information concept that may play an important role in holography and in understanding the physics of the black hole interior. We consider quantum computational complexity for $n$ qubits using…
Geometric phase (GP) independent of energy and time rely only on the geometry of state space. It has been argued to have potential fault tolerance and plays an important role in quantum information and quantum computation. We present the…
Several years ago the so-called quantum geometrodynamics in extended phase space was proposed. The main role in this version of quantum geometrodynamics is given to a wave function that carries information about geometry of the Universe as…
Superanalysis can be deformed with a fermionic star product into a Clifford calculus that is equivalent to geometric algebra. With this multivector formalism it is then possible to formulate Riemannian geometry and an inhomogeneous…
Geometric and holonomic quantum computation utilizes intrinsic geometric properties of quantum-mechanical state spaces to realize quantum logic gates. Since both geometric phases and quantum holonomies are global quantities depending only…
This paper reviews recent attempts to describe the two- and three-qubit Hilbert space geometries with the help of Hopf fibrations. In both cases, it is shown that the associated Hopf map is strongly sensitive to states entanglement content.…
Here is discussed application of the Weyl pair to construction of universal set of quantum gates for high-dimensional quantum system. An application of Lie algebras (Hamiltonians) for construction of universal gates is revisited first. It…