This page is dedicated to documenting the process and outcomes of the Honors Summer Mathematics Camp (HSMC) undergraduate Research Project I am conducting at Texas State University (online) during the summer. Visualization of Hopf fibration as a geometric structure of Bloch sphere in quantum computing
If you have any questions or inquiries about this project, please feel free to contact me at wvx17@txstate.edu.
Here is the description of the projects on summer 2025:
Project 1. Information Geometry of Quantum Computing (with Angela Yue, Jessie Wang)
Quantum computing can be studied through the lens of geometry, where the space of quantum states is naturally equipped with an information-geometric structure. Information geometry is a broad field that studies statistical models using differential geometry. This project explores these geometric foundations, referencing “The Geometry of Quantum Computing” by E. Ercolessi, R. Fioresi, and T. Weber.
The main goal is to investigate the relationship between entanglement entropy and quantum state geometry through concrete two-qubit quantum circuits. We aim to understand how geometric structures influence quantum information processing and develop an intuition for quantum state evolution in this framework.
We begin by introducing fundamental concepts, including qubits, density operators, and quantum logic gates, which will serve as the foundation for our study. Next, we examine key ideas from information geometry, such as the Fisher matrix, and explore their quantum counterparts, leading to the Quantum Geometric Tensor, a natural Kähler metric on the space of qubits.
This project is particularly suitable for undergraduate students interested in learning differential geometry at the graduate level in the future.
Problem Set 1: Linear Algebra and Matrix Foundations for Quantum States, (Angela Yue's work, Jessie Wang's work)
Problem Set 2: Curvature and Geometry on surfaces, (Angela Yue's work, Jessie Wang's work) recorded video
Problem set 3: Quotient Spaces and Quantum Geometry, recorded video, recorded video on more clarification on Hopf fibration
Project 2. ZX Algebra and Spider Fusion (with Evelyn Li, Jason Cheng)
In addition to the geometric approach, a separate but related project will be carried out focusing on ZX Algebra and the diagrammatic manipulation of quantum circuits.
This project studies how quantum operations can be simplified and analyzed through the ZX-Calculus, an elegant graphical language for reasoning about quantum states and gates. Students will investigate the Spider Fusion Law, Copy Law, and the equivalence between ZX-diagram rewriting and algebraic properties of projectors in the computational and Hadamard bases.
This project is independently structured but shares foundational topics with the geometric project, allowing cross-discussion and collaboration among participants.
Key topics include:
Problem Set 1: Linear Algebra and Matrix Foundations for Quantum computing (Evelyn Li’s work, Jason Cheng's work).
Problem Set 2: Standard ZX Calculus and Spider Law, (Evelyn Li's work, Jason Cheng's work), recorded video.
Problem Set 3: Quotient Space and Weighted Projective Line, (Evelyn Li's work, Jason Cheng's work), recorded video.