On the geometry of Variational Quantum Algorithms: Hopf fibrations, entanglement and the natural gradient descent - Habermann, Gustavo Schranck (2024)
Variational quantum algorithms were proposed as a strategy to obtain quantum advantage using the current state of the art quantum devices, the so called noisy intermediate scale quantum (NISQ) devices, by combining low-depth parameterized quantum circuits and classical optimization methods, such as the natural gradient descent. In this work we investigate geometrical aspects of variational quantum algorithms and the natural gradient descent in one and two qubits settings using tools of Riemannian geometry and information geometry, exploring the geometry of quantum states and the associated Hopf fibration structure as well as the behaviour of the natural gradient descent on statistical models contained in Riemannian manifolds. In addittion, recent results connecting entanglement, scalar curvature and two qubits parameterized quantum circuit performance are discussed and compared to our own calculations.