Geometric algebra and information geometry for quantum computational software
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Authors
Cafaro, Carlo
Issue Date
2017-03-15
Type
Article
Language
en_US
Keywords
geometric Clifford algebras , probability theory , quantum algorithms , quantum mechanics , Riemannian geometry , information geometry , Walsh–Hadamard operation
Alternative Title
Physica A: Statistical Mechanics and its Applications
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.
Description
Citation
Cafaro, C. (2017, March 15). Geometric algebra and information geometry for quantum computational software. Physica A: Statistical Mechanics and its Applications, 470, 154–196. doi:10.1016/j.physa.2016.11.117
Publisher
Physica A: Statistical Mechanics and its Applications
License
Journal
Volume
Issue
PubMed ID
DOI
ISSN
0378-4371