Research Interests:

I am mostly a quantum person, and have also worked on kinetic aspects of biological models and stochastic differential equations. 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 study quantum algorithms and their limitations for classical differential equations. Keywords: time-marching, 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. We also discuss the limitation of such quantum solvers.
• *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.
  Highlights:

  • Time-dependent Hamiltonian Simulation and Superconvergence: 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. The first work is qHOP and improved version is Magnus based quantum algorithm.
  • 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.