# Geometric Algebra and Information Geometry for Quantum Computational Software

## TL;DR

This paper employs geometric algebra and information geometry to analyze and enhance the understanding of Grover's quantum search algorithm, focusing on algebraic efficiency, geometric interpretation, and the transition from digital to analog descriptions.

## Contribution

It introduces a novel geometric algebra and information geometry framework for analyzing Grover's algorithm, linking discrete and continuous quantum descriptions and revealing new insights into quantum speedup.

## Key findings

- Enhanced algebraic description of Grover iterate using geometric algebra
- Interpretation of Grover's algorithm as a geodesic on a quantum state manifold
- Identification of the superfluity of Walsh-Hadamard operation from an information geometry perspective

## Abstract

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.