Knill introduced a generalization of stabilizer codes, in this note called Clifford codes. It remained unclear whether or not Clifford codes can be superior to stabilizer codes. We show that Clifford codes are stabilizer codes provided that the abstract error group is given by an extraspecial p-group. Suppose that the abstract error group has an abelian index group, then we show that a Clifford...

Clifford codes are a class of quantum error control codes that form a natural generalization of stabilizer codes. These codes were introduced in 1996 by Knill, but only a single Clifford code was known, which is not already a stabilizer code. We derive a necessary and sufficient condition that allows to decide when a Clifford code is a stabilizer code, and compile a table of all true Clifford c...

Clifford algebras have broad applications in science and engineering. The use of Clifford algebras can be further promoted in these fields by availability of computational tools that automate tedious routine calculations. We offer an extensive demonstration of the applications of Clifford algebras in electromagnetism using the geometric algebra G ≡ Cl3,0 as a computational model in the Maxima c...

The well-known classification of the Clifford algebrasCl(r, s) leads to canonical forms of complex and real representations which are essentially unique by virtue of the Wedderburn theorem. For s ≥ 1 representations of Cl(r, s) on R are obtained from representations on R by adding two new generators while in passing from a representation of Cl(p, 0) on R to a representation of Cl(r, 0) on R the...

Standard formulation is unable to distinguish between the (+ + +−) and (−−−+) spacetime metric signatures. However, the Clifford algebras associated with each are inequivalent, R(4) in the first case (real 4 by 4 matrices), H(2) in the latter (quaternionic 2 by 2). Multivector reformulations of Dirac theory by various authors look quite inequivalent pending the algebra assumed. It is not clear ...

We study reversible quantum cellular automata with the restriction that these are also Clifford operations. This means that tensor products of Pauli operators (or discrete Weyl operators) are mapped to tensor products of Pauli operators. Therefore Clifford quantum cellular automata are induced by symplectic cellular automata in phase space. We characterize these symplectic cellular automata and...

The representation of geometric objects and their transformation are the two key aspects in computer graphics applications. Traditionally, computer-intensive matrix calculations are involved in modeling and rendering three-dimensional (3D) scenery. Geometric algebra (aka Clifford algebra) is attracting attention as a natural way to model geometric facts and as a powerful analytical tool for sym...

One particular approach to quantum groups (matrix pseudo groups) provides the Manin quantum plane. Assuming an appropriate set of non-commuting variables spanning linearly a representation space one is able to show that the endomorphisms on that space preserving the non-commutative structure constitute a quantum group. The noncommutativity of these variables provide an example of non-commutativ...

Averaging measured data is an important issue in computer vision and robotics. Integrating the pose of an object measured with multiple cameras into a single mean pose is one such example. In many applications data does not belong to a vector space. Instead, data often belongs to a non–linear group manifold as it is the case for orientation data and the group of three–dimensional rotations SO(3...

Neural computation in Clifford algebras, which include familiar complex numbers and quaternions as special cases, has recently become an active research field. As always, neurons are the atoms of computation. The paper provides a general notion for the Hessian matrix of Clifford neurons of an arbitrary algebra. This new result on the dynamics of Clifford neurons then allows the computation of o...