Geometric Algebra in Quantum Computational Intelligence

Prof. Dr. Eng. Eduardo Bayro-Corrochano

Institute of Automation and Robotics of the Poznan University of Technology Poznan, Poland, eduardo.bayro@put.poznan.pl

 

With rapid advancements in AI and machine learning, coupled with significant progress in quantum computing, quantum machine learning is emerging as a revolutionary field. Quantum computers excel at solving complex problems that challenge classical systems, leveraging quantum algorithms for efficient computation. This progress unlocks vast opportunities across nearly all aspects of modern life.

In traditional quantum mechanics, the tensor product is used to construct multiparticle states and define operators acting on them, serving as a notational tool to separate the Hilbert spaces of different particles. Alternatively, the geometric algebra framework provides a unique representation of the tensor product through the geometric product, utilizing multivectors. Unlike tensor products, which lack an intuitive geometric interpretation, the geometric product and k-vectors (such as points, lines, planes, and volumes) offer a clear visualization. Moreover, entangled quantum states can be reinterpreted as k-vectors, representing structured collections of geometric shapes—including vectors, lines, planes, volumes, hyperplanes, and hypervolumes. For a deeper exploration of quantum theory within the geometric algebra framework, refer to E. Bayro-Corrochano’s Geometric Algebra Applications Vol. III: Integral Transforms, Machine Learning, and Quantum Computing (Springer Verlag, 2024).

Quantum machine learning is an evolving field that integrates quantum computing with machine learning methodologies. In this lecture, we will explore the core principles of quantum machine learning within the framework of geometric algebra. Furthermore, we will examine advanced algorithms, including the Quantum Quaternion Fourier Transform, Geometric Algebra Quantum Convolutional Neural Networks, and the Geometric Fuzzy Inference Engine for robotic decision-making.

 

Prof. Dr. Eng. Eduardo Bayro-Corrochano

Institute of Automation and Robotics of the Poznan University of Technology Poznan, Poland

eduardo.bayro@put.poznan.pl

 

Prof. Dr. Bayro-Corrochano is a distinguished leader and internationally recognized scientist and educator in Geometric Cybernetics. His expertise lies in applying Clifford geometric algebras across various fields, including pattern recognition, image processing, computer vision, artificial intelligence, neurocomputing, machine learning, control, robotics and quantum computing.

He has pioneered several advancements, including the geometric MLP, Clifford Support Vector Machines, quaternion quantum neural networks, and the innovative quaternion spike neural networks for pattern recognition and neuro control. Additionally, he developed the Quaternion Wavelet Transform and Quaternion Fourier Transform using space-time metrics, along with Quaternion Fractional FFT and Quaternion Quantum FFT for color image processing. His contributions also extend to the interpolation of geometric entities (such as lines, planes, circles, spheres, and hyperplanes) using motor algebra (SE(3)) and a Bézier approach in the Study manifold.

In the realm of geometric algebra, his groundbreaking work includes contributions to kinematics, dynamics, Euler-Lagrange and Newton-Euler recursive algorithms, port Hamiltonians, and the Koopman Operator for phase space computations. His research has significantly impacted nonlinear control, particularly in robot vision and general robotics. Furthermore, he is a strong advocate for geometric quantum computing within the Clifford geometric algebra framework.

 

His scientific contributions are particularly well represented in three of his eight books:

These books serve as valuable resources for graduate courses and provide inspiration for researchers and engineers working in cybernetics and related fields.

Prof. Bayro-Corrochano has served as an Associate Editor for the IEEE Transactions on Neural Networks and Learning Systems and the Journal of Mathematical Imaging and Vision. He is also a member of the editorial boards for the Journal of Pattern Recognition and Journal of Robotica. His achievements have been recognized with the First Prize in Science and Technology from the State of Jalisco, Mexico, in both 2003 and 2009. He is a Fellow of the International Association of Pattern Recognition Society and a senior member of IEEE.

He has played a key role in organizing major international conferences, serving as General Chair of ICPR 2016 (Dec. 4-8, Cancun, Mexico) and IEEE/RAS Humanoids 2016 (Nov. 15-17, Cancun, Mexico). He is also set to be the General Chair of IEEE/RAS Humanoids 2026 and IEEE/RAS ICRA 2028 both in Guadalajara, Mexico.