The well-known classification of the Clifford algebras $Cl(r,s)$ leads to canonical forms of complex and real representations which are essentially unique by virtue of the Wedderburn theorem. For $s\ge 1$ representations of $Cl(r,s)$ on $R^{2N}$ are obtained from representations on $R^N$ by adding two new generators while in passing from a representation of $Cl(p,0)$ on $R^N$ to a representation of $Cl(r,0)$ on $R^{2N}$ the number of generators that can be added is either 1, 2 or 4, according as the Clifford algebra represented on $R^N$ is of real, complex or quaternionic type. We have expressed canonical forms of these representations in terms of the complex and quaternionic structures in the half dimension and we obtained algorithms for transforming any given representation of $Cl(r,s)$ to a canonical form. Our algorithm for the transformation of the representations of $Cl(8d+c,0)$, $c\le 7$ to canonical forms is based on finding an abelian subalgebra of $Cl(8d+c,0)$ and its invariant subspace. Computer programs for determining explicitly the change of basis matrix for the transformation to canonical forms are given for lower dimensions. The construction of the change of basis matrices uniquely up to the commutant provides a constructive proof of the uniqueness properties of the representations and may have applications in computer graphics and robotics.
We propose to represent both $n$--qubits and quantum gates acting on them as elements in the complex Clifford algebra defined on a complex vector space of dimension $2n.$ In this framework, the Dirac formalism can be realized in straightforward way. We demonstrate its functionality by performing quantum computations with several well known examples of quantum gates. We also compare our approach with representations that use real geometric algebras.
Formulas to calculate multivector exponentials in a base-free representation and in a orthonormal basis are presented for an arbitrary Clifford geometric algebra Cl(p,q). The formulas are based on the analysis of roots of characteristic polynomial of a multivector. Elaborate examples how to use the formulas in practice are presented. The results may be useful in the quantum circuits or in the problems of analysis of evolution of the entangled quantum states.
We study the properties of a bosonization procedure based on Clifford algebra valued degrees of freedom, valid for spaces of any dimension. We present its interpretation in terms of fermions in presence of $\mathbb{Z}_2$ gauge fields satisfying a modified Gauss' law, resembling Chern-Simons-like theories. Our bosonization prescription involves constraints, which are interpreted as a flatness condition for the gauge field. Solution of the constraints is presented for toroidal geometries of dimension two. Duality between our model and $(d-1)$-form $\mathbb{Z}_2$ gauge theory is derived, which el…
