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 (Angela Yue's work), 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.
Here is the description of the projects on summer 2026:
Project 3. Quantum Cryptography from Algebraic Curves to Hardware Implementation (with Audrey Hendarto, Angela Wang, Gordon Chen, and Savanna Rocha)
Modern cryptography lies at the intersection of algebra, number theory, computer science, and hardware engineering. While many cryptographic systems rely on difficult mathematical problems such as integer factorization or discrete logarithms, the emergence of quantum computing introduces new computational capabilities that challenge the security assumptions of classical cryptography.
This project explores quantum cryptography from both the mathematical and engineering perspectives. We begin with the algebraic foundations underlying modern cryptographic systems, including finite fields, elliptic curves, and algebraic curves. These structures provide the basis for many widely used cryptographic protocols and offer a rich geometric viewpoint for understanding secure communication.
The project then introduces quantum algorithms that threaten classical cryptographic schemes. In particular, we study Grover’s Algorithm, which provides a quadratic speedup for brute-force key search, and analyze its implications for symmetric cryptography such as AES-128. Through explicit examples, students will learn how quantum search operates on a superposition of candidate keys and how repeated Grover iterations amplify the probability of finding the correct secret key.
A central component of the project is the implementation of AES-128 as a reversible Boolean circuit suitable for quantum computation. Students will investigate how classical hardware descriptions of AES components—including XOR operations, S-boxes, ShiftRows, and MixColumns—can be translated into reversible logic and quantum gate constructions. This provides a concrete bridge between modern cryptography, quantum algorithms, and hardware realization.
Finally, students will explore FPGA-based implementations of AES-128 and compare classical hardware costs with the resources required for quantum attacks. This hardware-oriented perspective highlights the practical challenges of implementing cryptographic systems and the role of specialized architectures in secure computing.
This project is particularly suitable for students interested in algebra, cryptography, quantum computing, FPGA design, computer engineering, and mathematical aspects of cybersecurity.
Key topics include:
Note 1: Finite Fields, Algebraic Curves, and Foundations of Modern Cryptography
Note 3: Hardware Realization of Quantum Cryptographic Attacks
Final Presentations
Project 4. Topological Quantum Error Correction: Surface Codes and Quantum LDPC Codes (with Rhea Ghosal, Isaiah Bae, and Dillan Garner)
Quantum computers possess remarkable computational power, but quantum information is extremely fragile. Even small interactions with the environment can introduce errors that rapidly destroy quantum states. Unlike classical computers, where information can be protected through simple redundancy, quantum systems are constrained by the no-cloning theorem, making error correction significantly more challenging.
This project studies how geometry, topology, and graph theory can be used to protect quantum information from noise. We begin with the fundamental ideas of classical error-correcting codes and gradually build toward modern quantum error correction schemes. Particular emphasis is placed on Surface Codes and Quantum Low-Density Parity Check (qLDPC) Codes, two of the most important architectures currently being developed for large-scale fault-tolerant quantum computation.
The first part of the project introduces stabilizer codes, parity-check matrices, and syndrome measurements. Students will learn how quantum errors can be detected indirectly through measurements that preserve the encoded quantum information. We then study surface codes, where qubits are arranged on a two-dimensional lattice and logical information is encoded using topological structures such as loops and homology classes.
The second part of the project investigates the relationship between topology and quantum information. Students will explore how surfaces, graphs, Euler characteristics, and homology groups naturally appear in the construction of quantum codes. These ideas reveal a surprising connection between algebraic topology and practical quantum engineering.
The final stage introduces modern Quantum LDPC Codes, a rapidly developing area of research that seeks to reduce the enormous hardware overhead required by traditional surface-code architectures. Students will study Tanner graphs, sparse parity-check matrices, expander graphs, and recent breakthroughs in quantum LDPC constructions that may significantly accelerate the development of scalable quantum computers.
This project is particularly suitable for students interested in topology, algebra, geometry, graph theory, coding theory, cryptography, and quantum computing.
Key topics include: