Research Interests:

I am mostly a quantum person, and have also worked on some biological models and stochastic differential equations. After joining Berkeley and attending the Simons Institute program on the Quantum Wave in Computing, I became intrigued by quantum algorithms. At this point, my passion mainly lies in the following three different types of quantum problems:

  • classical for quantum, namely to simulate and analyze quantum systems via classical algorithms. Keywords: nonadiabatic dynamics, ab initio molecular dynamics, surface hopping. (click to see more) My expertise is to deal with the non-adiabaticity (beyond Born-Oppenheimer approximation) and/or nonlinearity of the Schrodinger equations emerging from the ab initio molecular dynamics.
    • Mathematical Justification of Surface Hopping: Trajectory surface hopping is one of the most popular algorithms in simulating quantum nonadiabatic dynamics. Its mathematical justification, however, was not well-understood, especially in diabatic representation. Our sequential work provide the mathematical justification of surface hopping algorithm in diabatic representation, and further asymptotic analysis in both Marcus (perturbative) and non-perturbative regimes that match the Marcus golden scaling and draw the connection with Ehrenfest dynamics.
    • Observable error bounds for mean-field dynamics: For quantum-classical molecular dynamics, our work (with my undergraduate student) provides observable error bounds that are uniform in the rescaled Planck constant. No such result is previously known for nonlinear systems; For Ehrenfest dynamics, our work proves the first existence of a global in-time meshing strategy independent of rescaled Planck constant for physical observables associated to a nonlinear Schrodinger-type system.
    • TDDFT for metallic systems: Our work proposes the framework of the Parallel transport (PT) dynamics for general (mixed) quantum states. Equivalent to the Schrodinger dynamics up to a gauge choice, PT dynamics allows a significantly larger time-step and our new commutator-type error bound justifies its efficiency for the nonlinear Hamiltonians and beyond the near-adiabatic regimes.
  • quantum for classical, namely to design quantum algorithms for classical differential equations. Keywords: time-marching, Carleman linearization, high dimensionality (click to see more) Recent advances of quantum algorithms provide a new viewpoint of overcoming the curse of dimensionality. However, due to no-cloning theorem, quantum measurements and so on, the major numerical challenges here can be very different from numerical analysis of classical algorithms. This area is still in its early stage. Together with collaborators, we are exploring slowly and see what we can do.
    • Time-marching: The time-marching strategy, which propagates the solution from one time step to the next, is a natural strategy for solving time-dependent differential equations adopted by nearly all algorithms on classical computers, as well as for solving the Hamiltonian simulation problem on quantum computers. However, for more general linear differential equations, a time-marching based quantum solver has been considered impractical, as it can suffer from exponentially vanishing success probability with respect to the number of time steps. Our work solves this problem by repeatedly invoking a technique called the uniform singular value amplification, and the overall success probability can be lower bounded by a quantity that is independent of the number of time steps. The proposed time-marching strategy can be paired with any reasonable short-time numerical integrator. The complexity of the algorithm depends linearly on the amplification ratio, which quantifies the deviation from a unitary dynamics. We show that when paired with truncated Dyson series, the Q dependence attains query complexity lower bound. Our analysis also raises some open questions concerning the differences between time-marching and QLSA based methods for solving differential equations.
    • Together with Maryland and Microsoft Quantum researchers, we propose an efficient quantum algorithm for nonlinear reaction-diffusion equations that extends the Carleman linearization approach. With improved error analysis, it survives under the grid refinement. Output for energy estimation is analyzed.
  • *quantum for quantum* (major focus), namely to study quantum algorithms for quantum systems and quantum problems.  Keywords: Hamiltonian simulation, unbounded operators, observable error bounds (click to see more) I am interested in Hamiltonian simulation involving unbounded operators, including the development of quantum algorithms and proof of error bounds for such problems.
    • Quantum highly oscillatory protocol (qHOP): a first quantum algorithm for Hamiltonian simulation that is both insensitive to the rapid changes of the time-dependent Hamiltonian and exhibits commutator scaling. Interestingly, the algorithm is proved to achieve a surprising superconvergence for Schrodinger equation.
    • Vector norm error analysis, a framework for error estimate of quantum algorithms for unbounded Hamiltonian simulation that drastically reduces the overhead caused by number of spatial grids when taking into account the information of the initial state. In this sense, our result outperforms all previous error bounds in the quantum simulation literature.
  • The tools and techniques that I use the most, but not limited to, are numerical analysis, semiclassical calculus, stochastic analysis and quantum computation. I am by no means an expert in all these areas, but as a problem-targeted person, I learn the techniques along the way to solve the practical problems in mind.

Publication and Preprints:

Recent and Upcoming Talks:

(not up to date; talks since Dec 2022; for a full list, please see my cv - last updated @ April 11 2023.)
  • 2023 Jul, 17th Annual Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023), University of Aveiro, Portugal
  • 2023 Apr, CIFAR’s Quantum Information Science (QIS) Research Program Meeting, Boston
  • 2023 Apr, Applied Math Seminar, Stanford University
  • 2023 Apr, Computational and Applied Mathematics Colloquium, Pennsylvania State University
  • 2023 March, the annual American Physical Society (APS) March Meeting 2022 (invited address on quantum computing algorithms), Las Vegas, Nevada
  • 2023 Feb, Kavli ITS Young Scholars Forum 2023, Kavili Institute for Theoretical Sciences
  • 2023 Feb, Departmental Seminar, Department of Mathematics, Massachusetts Institute of Technology
  • 2023 Jan, Applied Math Seminar, Courant Institute of Mathematical Sciences, New York University
  • 2023 Jan, Colloquium, Department of Mathematics, Duke University
  • 2023 Jan, Colloquium, Department of Mathematics, North Carolina State University
  • 2023 Jan, Colloquium, Department of Mathematics, Carnegie Mellon University
  • 2023 Jan, Colloquium in Applied Mathematics, University of Chicago
  • 2023 Jan, Applied Math Colloquium, University of California, Los Angeles
  • 2022 Dec, Departmental Seminar, Division of Applied Mathematics, Brown University
  • 2022 Dec, Mathematics Department Colloquium, University of Illinois Urbana-Champaign
  • 2022 Dec, Colloquium, Department of Computational Applied Mathematics and Operations Research, Rice University
  • 2022 Dec, Colloquium, School of Mathematics, Georgia Institute of Technology
  • 2022 Dec, Colloquium, Department of Mathematics, University of Wisconsin-Madison
  • 2022 Dec, Colloquium, School of Mathematics, University of Minnesota