A peculiarity of the geometry of the euclidean 3-sphere S is that it allows for the existence of compact without boundary minimally immersed surfaces. Despite a wealthy of examples of such surfaces, the only known tori minimally embedded in S are the ones congruent to the Clifford torus. In 1970 Lawson conjectured that the Clifford torus is, up to congruences, the only torus minimally embedded ...

Exact synthesis is a tool used in algorithms for approximating an arbitrary qubit unitary with a sequence of quantum gates from some finite set. These approximation algorithms find asymptotically optimal approximations in probabilistic polynomial time, in some cases even finding the optimal solution in probabilistic polynomial time given access to an oracle for factoring integers. In this paper...

Extended Clifford circuits straddle the boundary between classical and quantum computational power. Whether such circuits are efficiently classically simulable seems to depend delicately on the ingredients of the circuits. While some combinations of ingredients lead to efficiently classically simulable circuits, other combinations that might just be slightly different, lead to circuits which ar...

The generalized Pauli group and its normalizer, the Clifford group, have a rich mathematical structure which is relevant to the problem of constructing symmetric informationally complete POVMs (SIC-POVMs). To date, almost every known SIC-POVM fiducial vector is an eigenstate of a “canonical” unitary in the Clifford group. I show that every canonical unitary in prime dimensions p > 3 lies in the...

I review the equivalence between duality operators on two-forms and conformal structures in four dimensions, from a Clifford algebra point of view (due to Urbantke and Harnett). I also review an application, which leads to a set of ”neighbours” of Einstein’s equations. An attempt to formulate reality conditions for the ”neighbours” is discussed. Talk presented at the second Max Born Symposium o...

We study smooth maps from an open subset of a real projective space to a sphere such that all germs of straight lines go to germs of circles (or points). We give a construction of such maps based on representations of Clifford algebras. It provides a complete purely geometric description of Clifford algebras representations in terms of circles. We also describe a connection between our problem ...

This paper presents a geometric neurocomputing approach for 3D pose recognition using the framework of the geometric Clifford algebras. The type of geometric problems like pattern recognition and 3D pose recognition can be very efficiently handled using geometric neural networks. Our experimental part shows the application of generalized Clifford moments for pattern recognition and 3D pose esti...

This paper is a study of solutions to nonlinear Dirac equations, in domains in Euclidean space, which are generalizations of the Clifford Laplacian as well as elliptic equations in divergence form. A Caccioppoli estimate is used to prove a global integrability theorem for the image of a solution under the Euclidean Dirac operator. Oscillation spaces for Clifford valued functions are used which ...

This paper is concerned with quaternion-valued functions on the plane and operators which act on such functions. In particular, we investigate the space L(R,H) of square-integrable quaternion-valued functions on the plane and apply the recently developed Clifford-Fourier transform and associated convolution theorem to characterise the closed translation-invariant submodules of L(R,H) and its bo...