Projects

Here are some project write-ups, completed in collaboration with amazing peers. While the results may not have been novel enough for publication, I hope they can still offer valuable insights by:

• Highlighting directions that didn’t work out, and identifying key challenges.
• Numerically verifying and confirming established results.
• Demonstrating the application of elegant theorems and algorithms to similar problems and scenarios.

Differentially Private SGD with Curriculum Learning

• We incorporated curriculum learning into stochastic gradient descent with differential privacy to better balance data privacy and model performance.
• We experimented different noise injection schedules (noise curriculum) and re-arranged sample ordering (sample curriculum) when training a deep learning model.
• Results show promising accuracy improvements with noise curriculum, but little evidence of performance gains from sample curriculum.

Differentiable Pulse-Based Variational Quantum Eigensolver (PPT)

• We proposed a polynomial parameterization method for pulse-based VQE, reducing parameters from $M \times Q$ to $n \times Q$ (where $M$ is the number of time segments and $Q$ is the number of qubits), along with an efficient analytic gradient formula that lowers computational cost to $O(1)$ compared to $O(M)$ in finite difference methods, though it applies to a limited set of pulse sequences.
• Implementations confirm the proposed pulse-based VQE is fast, but we cannot reach the theoretical minimum due to pulse sequence ansatz choices, which is limited by our gradient calculation method.

Quantum Dynamic Programming

• We applied the divide-and-conquer dynamic programming approach ABIKPV18 to NP-hard problems like graph coloring, minimum clique cover, and minimum dominating set, achieving time complexity improvements from $O(2.4423^n)$ classically to $O(1.9140^n)$ quantumly on certain instances.
• We highlighted that for problems like the dominating set and minimum vertex cover, dynamic programming underperforms compared to direct application of Grover’s algorithm on classical brute-force methods, suggesting the need for new approaches to utilize dynamic programming effectively.

Towards Physically-Consistent, Chaotic Spatiotemporal Dynamics with Echo State Networks

• This study investigates the use of echo state networks (ESNs) for time-series forecasting in chaotic physics.
• We compare a basic ESN with two physics-informed variants on the Lorenz attractor and then test the ESN on a large-scale atmospheric model and a real-world weather dataset.
• Results show that a well-tuned traditional ESN can outperform physics-informed methods. While the ESN accurately predicts the global evolution of atmospheric primitive equations over short periods (~67 hours), it struggles with real-world data.
• Accepted to AAAI Spring Symposium Series 2021

Investigation of Area Law for Local Hamiltonians

• We implemented generic and RRG algorithms to find ground states of local Hamiltonians, including Heisenberg and AKLT models.
• We verified an area law for von Neumann entanglement entropy in the XXZ model for up to $14$ qubits.
• We confirmed exponential Schmidt coefficient distribution up to 13 qubits and predicted the slope for $14$ qubits, with discrepancies within $0.1$.