  Alternative mathematical explorations in quantum computing can be of great
scientific interest, especially if they come with penetrating physical
insights. In this paper, we present a critical revisitation of our geometric
(Clifford) algebras (GAs) application in quantum computing as originally
presented in [C. Cafaro and S. Mancini, Adv. Appl. Clifford Algebras 21, 493
(2011)]. Our focus is on testing the usefulness of geometric algebras (GAs)
techniques in two applications to quantum computing. First, making use of the
geometric algebra of a relativistic configuration space (a.k.a., multiparticle
spacetime algebra or MSTA), we offer an explicit algebraic characterization of
one- and two-qubit quantum states together with a MSTA description of one- and
two-qubit quantum computational gates. In this first application, we devote
special attention to the concept of entanglement, focusing on entangled quantum
states and two-qubit entangling quantum gates. Second, exploiting the
previously mentioned MSTA characterization together with the GA depiction of
the Lie algebras SO(3;R) and SU(2;C) depending on the rotor group formalism, we
focus our attention to the concept of universality in quantum computing by
reevaluating Boykin's proof on the identification of a suitable set of
universal quantum gates. At the end of our mathematical exploration, we arrive
at two main conclusions. Firstly, the MSTA perspective leads to a powerful
conceptual unification between quantum states and quantum operators. More
specifically, the complex qubit space and the complex space of unitary
operators acting on them merge in a single multivectorial real space. Secondly,
the GA viewpoint on rotations based on the rotor group carries both conceptual
and computational upper hands compared to conventional vectorial and matricial
methods.
