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. This is also the first algorithm for quantum differential equation that achieves optimal state-preparation cost.
  • Qubit-efficient Quantum DE solver with theoretical gaurantee: As quantum hardware rapidly advances toward the early fault-tolerant era, a key challenge is to develop quantum algorithms that are not only theoretically sound but also hardware-friendly on near-term devices. Our work addresses this challenge by designing a quantum differential equation solver that is hardware-friendly and with theoretical gaurantee. More background: Quantum algorithms can be generally classified into two broad categories: heuristic algorithms, such as variational quantum algorithms, and fault-tolerant algorithms with provable performance guarantees. Heuristic algorithms are often considered near-term feasible, but it is typically challenging to establish rigorous guarantees for them. In contrast, fault-tolerant algorithms are designed with theoretical guarantees in mind, often achieving excellent asymptotic scaling. However, their practical implementation is usually beyond the capability of near-term devices due to the reliance on advanced quantum techniques -- such as those requiring large numbers of ancilla qubits and complex control circuits -- which remain out of reach for current hardware. One expection is Trotterization for Hamiltonian simulation, which is a fault-tolerant algorithm with provable guarantees but it is so simple that can be implemented on the current hardware. However, the nice features of Trotterization (no ancillatary etc) is mainly due to its application to unitary dynamics (Hamiltonian simulation). When it is applied to non-unitary dynamics such as the general linear differential equation whose solution norm is not perserved, due to the non-unitary nature, some form of block-encoding is needed for each time step, which will result in an exponentially vanishing success probablity in the number of time steps. To remedy it, one needs to either ultilize some less near-term techniques typically comes with a lot more ancillatory qubits and control units, such as QLSA, compression gadget in time-marching, and LCU in linear combination of Hamiltonian simulation. In this work, we want to address the question: Can we design a quantum algorithm that is both near-term (without relying on advanced fault-tolerant subroutines) and with provable guarantees for linear differential equations? We provide a first algorithm in this category.
• *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 with a first-order convergence and the improved version is Magnus based quantum algorithm that is of arbitrary order. Interestingly, despite the existence of p! terms in p-th order Magnus expansion, we can design a quantum algorithm with only poly(p) cost (see our work on high-order Magnus expansion and circuit constructions). Also for p-th order quantum Magnus algorithm, we observed a suprising 2p superconvergence in the digital simulation of Schrodinger PDE (unbounded Hamiltonian simulation problem). The proof relies on nontrivial arguments inspired by semiclassical and discrete microlocal analysis (see continuous case and fully discrete superconvergence.
  • 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.
  • Similarly, Observable error bounds that quantifies the error in terms of specific observables can typically reduce the cost estimate. While constant factor improvements are always expected and not surprising. In my works, I estabilish asymptotic scaling improvements for a number of scenarios. （1) Example 1: Semiclassical Schrodinger equation (with the Planck constant h << 1) is an important multiscale quantum dynamical system arising from molecular dynamcis, as carefully reviewed in the two acta numerica reviews (1 and 2). The main difficulty is that the cost for all algorithms in the worst case is expected to be poly(1/h) to properly capture the wavefunctions. However, it has been numerically observed in 2002 that for a class of observables, one can make the time step independent on h. Since then, great applications and significant development in the semiclassical limiting arguments (e.g., Wigner measures, husimi functions, etc) have happened in subsequent years. Such numerical analysis results typically work with spatially continuous case, and is at most additive instead of uniform -- due to the estimate strategy using the limiting dynamics as a stepping stone. This would impose a severe restriction on the desired accuracy. Our result work provides the first uniform in h error bound for Trotterized observables in fully discrete setting without sacrificing the convergence order of the algorithm. Unlike previous strategies that working with the limiting macroscopic dynamics as an intermediate step, we directly estimate the microsopic error. Our results naturally extends to a fully discrete case which provides a complete justification to the numerical observations. (2) Example 2: Such specific case error analysis has also found applications in quantum learning tasks, contributing to the first optimal scaling (Heisenberg-limited) Hamiltonian learning algorithm for many-body quantum systems (see the collaboration work here).
  • Quantum Simulation of Many-body Quantum Dynamics with Coulomb Potentials: One central question in many-body simulation and quantum computing is: Can one demonstrate that a (quantum) algorithm can efficiently simulate many-body quantum systems with a cost that scales polynomially with the system size? A physically fundamental and notoriously challenging system is many-body quantum systems with Coulomb interactions. These systems are the first-principle description for electronic and molecular dynamics. The difficulty is threefold: the Hilbert space dimension grows exponentially with particle number; both kinetic and potential operators are unbounded; and the Coulomb potential is singular and non-smooth, violating the regularity assumptions of prior many-body Trotter error analyses. In this work, we close this gap by rigorously quantifying the Trotter error for many-body Coulomb Hamiltonians. We prove an optimal 1/4-order convergence rate with explicit polynomial dependence on system size, holding for all initial wavefunctions in the domain of the Hamiltonian. Optimality is certified by prior results showing that the 1/4 rate can be saturated by a specific ground state. To the best of our knowledge, this is the first result to achieve this sharp convergence rate--even in the one-body setting--and also the first to establish system-size dependence while treating the Coulomb potential as an unbounded operator. Our analysis treats the Coulomb potential as an unbounded operator without modification or regularization, and does not rely on spatial discretization, making it compatible with both first- and second-quantized circuit constructions.
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.