  The art of quantum algorithm design is highly nontrivial. Grover's search
algorithm constitutes a masterpiece of quantum computational software. In this
article, we use methods of geometric algebra (GA) and information geometry (IG)
to enhance the algebraic efficiency and the geometrical significance of the
digital and analog representations of Grover's algorithm, respectively.
Specifically, GA is used to describe the Grover iterate and the discretized
iterative procedure that exploits quantum interference to amplify the
probability amplitude of the target-state before measuring the query register.
The transition from digital to analog descriptions occurs via Stone's theorem
which relates the (unitary) Grover iterate to a suitable (Hermitian)
Hamiltonian that controls Schrodinger's quantum mechanical evolution of a
quantum state towards the target state. Once the discrete-to-continuos
transition is completed, IG is used to interpret Grover's iterative procedure
as a geodesic path on the manifold of the parametric density operators of pure
quantum states constructed from the continuous approximation of the parametric
quantum output state in Grover's algorithm. Finally, we discuss the
dissipationless nature of quantum computing, recover the quadratic speedup
relation, and identify the superfluity of the Walsh-Hadamard operation from an
IG perspective with emphasis on statistical mechanical considerations.
