Quantum Information Geometry and Quantum Estimation

Masahito Hayashi

Demetris P. K. Ghikas

Demetris P. K. Ghikas

Keiji Matsumoto

It is pointed out that Berry-Uhlmann’s parallelism plays key role in the first order asymptotic theory of quantum statistical estimation, and that the parallelism is closely related to Nagaoka’s quantum information geometry, which is already successfully applied to quantum estimation theory. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Mohammad Aamir Sohail, Mohsen Heidari, S. Sandeep Pradhan

Variational quantum algorithms, optimized using gradient-based methods, often exhibit sub-optimal convergence performance due to their dependence on Euclidean geometry. Quantum natural gradient descent (QNGD) is a more efficient method that incorporates the geometry of the state space via a quantum information metric. However, QNGD is computationally intensive and suffers from high sample complexity. In this work, we formulate a novel quantum information metric and construct an unbiased estimator for this metric using single-shot measurements. We develop a quantum optimization algorithm that leverages the geometry of the state space via this estimator while avoiding full-state tomography, as in conventional techniques. We provide the convergence analysis of the algorithm under mild conditions. Furthermore, we provide experimental results that demonstrate the better sample complexity and faster convergence of our algorithm compared to the state-of-the-art approaches. Our results illustrate the algorithm’s ability to avoid saddle points and local minima.

Kimikazu Kato

In quantum information theory, a geometric approach, known as "quantum information geometry," has been considered as a powerful method. In this thesis, we give a computational geometric interpretation to the geometric structure of a quantum system. Especially we introduce the concept of the Voronoi diagram and the smallest enclosing ball problem to the space of quantum states. With those tools in computational geometry, we analyze the adjacency structure of a point set in the quantum state space. Additionally, as an application, we show an effective method to compute the capacity of a quantum channel.