Research
My current research can be categorized into two main directions:
Theoretical analysis of high-dimension problems or algorithms: I use PDE analysis as a toolbox to study the evolution of high-dimensional systems. The PDE tools, including gradient flow equation, and mean-field analysis, are helpful in formulating the evolution of a complicated high-dimensional system into certain mathematical descriptions (e.g. PDE) that are easier to handle.
In this direction, I mainly focus on analyzing modern machine learning algorithms, including Bayesian sampling methods, and over-parameterized neural networks.
Numerical simulation of high-dimension problems: Numerical simulation of a high-dimensional system is always a challenging problem because of the curse of dimensionality. For general high-dimensional systems, both the computational cost and the storage memory of the classical simulation grow exponentially fast in dimension. Thus, new methods and technics need to be developed for the simulation of high-dimensional systems.
My current interest in this direction lies in two different perspectives. 1. Many high-dimensional systems have their own structures, such as mean-field limit and low-rank structures in PDE. Using the structure information, it's possible to develop new methods that have much smaller complexity than the classical methods. 2. More recently, the development of quantum computing algorithms provides a different way to overcome the curse of dimensionality. Using the properties of quantum physics to store data and perform computations, a quantum computer has the potential to exponentially reduce computational cost and storage memory. On the other hand, the special structure of quantum computing also asks for a very different way to design the algorithm, which is always highly non-trivial.
Publications:
Preprints:
[10] Z. Ding, B. Li, L. Lin, Efficient quantum Gibbs samplers with Kubo-Martin-Schwinger detailed balance condition, arXiv/2404.05998, 2024.
[9] Z. Ding, E. N. Epperly, L. Lin, R. Zhang, The ESPRIT algorithm under high noise: Optimal error scaling and noisy super-resolution, arXiv/2404.03885, 2024.
[8] Z. Ding, H. Li, L. Lin, H. Ni, L. Ying, R. Zhang, Quantum Multiple Eigenvalue Gaussian filtered Search: an efficient and versatile quantum phase estimation method, arXiv/2402.01013, 2024.
[7] S. Chen, Z. Ding, Q. Li, Bayesian sampling using interacting particles, arXiv/2401.13100, 2024.
[6] Z. Ding, X. Li, L. Lin, Simulating open quantum systems using hamiltonian simulations, PRX Quantum, accepted, 2024.
[5] Z. Ding, T. Ko, J. Hao, L. Lin, X. Li, Random coordinate descent: a simple alternative for optimizing parameterized quantum circuits, arXiv/2311.00088, 2023.
[4] Z. Ding, C. Chen, L. Lin, Single-ancilla ground state preparation via Lindbladians, arXiv/2308.15676, 2023.
[3] Z. Ding, Y. Dong, Y. Tong, L. Lin, Robust ground-state energy estimation under depolarizing noise, arXiv/2307.11257, 2023.
[2] S. Chen, Z. Ding, Q. Li, S. Wright, On optimal bases for multiscale PDEs and Bayesian homogenization, arXiv/2305.12303, 2023.
[1] N. Abrahamsen, Z. Ding, G. Goldshlager, L. Lin, Convergence of stochastic gradient descent on parameterized sphere with applications to variational Monte Carlo simulation, arXiv/2303.11602, 2023.
Peer reviewed papers: