# Registered Data

## [00496] Recent development in Quantum Simulation and Stochastic Methods

**Session Date & Time**:- 00496 (1/3) : 3C (Aug.23, 13:20-15:00)
- 00496 (2/3) : 3D (Aug.23, 15:30-17:10)
- 00496 (3/3) : 3E (Aug.23, 17:40-19:20)

**Type**: Proposal of Minisymposium**Abstract**: This mini-symposium aims to bring together mathematicians and scientists working on quantum simulation and related topics to exchange ideas and share recent results. It highlights how the recent developments in computational tools, such as stochastic methods, quantum computing, fast algorithms, etc., make the simulation of large-scale or multiscale quantum systems feasible and thus expand the scope of quantum simulation. It also serves as a platform for presenting challenging quantum problems, which still call for novel methodologies to alleviate simulation costs.**Organizer(s)**: Lihui Chai, Zhiwen Zhang, Zhennan Zhou**Classification**:__35Q40__,__81-08__,__68Q12__,__65C20__**Speakers Info**:**Zhennan Zhou**(Peking University)- Guillaume Bal (University of Chicago)
- Jingrun Chen (University of Science and Technology of China)
- Di Fang (UC-Berkeley)
- Weiguo Gao (Fudan University)
- Tongyang Li (Peking University)
- Yingzhou Li (Fudan University)
- Zhongyi Huang (Tsinghua University)
- Yong Zhang (Tianjin University)
- Xiaofei Zhao (Wuhan University)

**Talks in Minisymposium**:**[01869] Quantum Orbital Minimization Method for Excited States Calculation on Quantum Computer****Author(s)**:**Yingzhou Li**(Fudan University)

**Abstract**: We propose a quantum-classical hybrid variational algorithm, the quantum orbital minimization method (qOMM), for obtaining the ground state and low-lying excited states of a Hermitian operator. Given parametrized ansatz circuits representing eigenstates, qOMM implements quantum circuits to represent the objective function in the orbital minimization method and adopts a classical optimizer to minimize the objective function with respect to the parameters in ansatz circuits. The objective function has an orthogonality constraint implicitly embedded, which allows qOMM to apply a different ansatz circuit to each input reference state. We carry out numerical simulations that seek to find excited states of H2, LiH, and a toy model consisting of four hydrogen atoms arranged in a square lattice in the STO3G basis with UCCSD ansatz circuits. Comparing the numerical results with existing excited states methods, qOMM is less prone to getting stuck in local minima and can achieve convergence with more shallow ansatz circuits.

**[01872] Frozen Gaussian Sampling for mixed quantum-classical dynamics****Author(s)**:**Zhennan Zhou**(Peking University)- Zhen Huang ( University of California, Berkeley)
- Limin Xu ( Tsinghua University)

**Abstract**: In this article, we propose a Frozen Gaussian Sampling (FGS) algorithm for simulating nonadiabatic quantum dynamics at metal surfaces with a continuous spectrum. This method consists of a Monte-Carlo algorithm for sampling the initial wave packets on the phase space and a surface-hopping type stochastic time propagation scheme for the wave packets. We prove that to reach a certain accuracy threshold, the sample size required is independent of both the semiclassical parameter and the number of metal orbitals, which makes it one of the most promising methods to study nonadiabatic dynamics. The algorithm and its convergence properties are also validated numerically. Furthermore, we carry out numerical experiments including exploring the nuclei dynamics, electron transfer, and finite-temperature effects, and demonstrate that our method captures the physics which can not be captured by classical surface hopping trajectories.

**[02028] On Quantum Speedups for Nonconvex Optimization via Quantum Tunneling Walks****Author(s)**:- Yizhou Liu (MIT)
- Weijie J. Su (University of Pennsylvania)
**Tongyang Li**(Peking University)

**Abstract**: Classical algorithms are often not effective for solving nonconvex optimization problems where local minima are separated by high barriers. In this paper, we explore possible quantum speedups for nonconvex optimization by leveraging the global effect of quantum tunneling. Specifically, we introduce a quantum algorithm termed the quantum tunneling walk (QTW) and apply it to nonconvex problems where local minima are approximately global minima. We show that QTW achieves quantum speedup over classical stochastic gradient descents (SGD) when the barriers between different local minima are high but thin and the minima are flat. Based on this observation, we construct a specific double-well landscape, where classical algorithms cannot efficiently hit one target well knowing the other well but QTW can when given proper initial states near the known well. Finally, we corroborate our findings with numerical experiments.

**[02280] The Random Feature Method for Time-dependent Problems****Author(s)**:- Jingrun Chen (University of Science and Technology of China)
- Weinan E (Peking University)
**Yixin Luo**(Suzhou Institute for Advanced Research, University of Science and Technology of China)

**Abstract**: We propose to solve time-dependent partial differential equations in the framework of random feature method. The approximate solution is constructed using space-time partition of unity and random feature functions, and is proved to enjoy the universal approximation property. The resulting least-squares problem can be solved using two different strategies, and the corresponding error estimates are provided.

**[02283] Bloch decomposition based method for Schroedinger equation with random inputs****Author(s)**:**Zhongyi Huang**(Tsinghua University)

**Abstract**: In this talk, we focus on the analysis and numerical methods for the Schroedinger equation with lattice potential and random inputs. Here we recall the well-known Bloch decomposition-based split-step pseudo-spectral method where we diagonalize the periodic part of the Hamilton operator so that the effects from dispersion and periodic lattice potential are computed together. Meanwhile, for the random non-periodic external potential, we utilize the generalize polynomial chaos with Galerkin procedure to form an ODE system which can be solved analytically. Furthermore, we analyse the convergence theory of the stochastic collocation method for the linear Schroedinger equation with random inputs. We provide sufficient conditions on the random potential and initial data to ensure the spectral convergence.

**[02728] Numerical methods for disordered NLS****Author(s)**:**Xiaofei Zhao**(Wuhan University)

**Abstract**: In this talk, I will consider the numerical solution for the disordered nonlinear Schrodinger equation (NLS). The model is a standard cubic NLS with a spatial random potential. The roughness and/or randomness of the potential brings some numerical difficulties. The performance of some classical time integrators will reviewed, and the low-regularity integrator will be applied to tackle the possible roughness. Then, a quasi-Monte Carlo time-splitting method will be considered to tackle the randomness and improve the sampling accuracy. The full error bound will be presented and numerically verified.

**[02786] Orthogonalization and Orthogonalization-free Algorithms****Author(s)**:**Weiguo Gao**(Fudan University)

**Abstract**: This talk consists of two parts. Orthogonalization algorithms are discussed in the first part. And we extend the error analysis result for shiftedCholeskyQR3 algorithm in oblique inner product and show that the new error bound is optimal. We also discuss the loss of orthogonality and verify our conclusions through numerical experiments. This is joint work with Rentao Xu. In the second part, orthogonalization-free algorithms are proposed to solve extreme eigenvalue problems. These algorithms achieve eigenvectors instead of eigenspace. Global convergence and local linear convergence are discussed. Efficiency of new algorithms are demonstrated on random matrices and matrices from computational chemistry. This is joint work with Yingzhou Li and Bichen Lu.

**[02812] Asymmetric transport computations in Dirac models of topological insulators****Author(s)**:**Zhongjian Wang**- Guillaume Bal (University of Chicago)
- Jeremy Hoskins (University of Chicago)

**Abstract**: We will present a fast algorithm for computing transport properties of two-dimensional Dirac operators with linear domain walls, which model the macroscopic behavior of the robust and asymmetric transport observed at an interface separating two two-dimensional topological insulators. Our method is based on reformulating the partial differential equation as a corresponding volume integral equation, which we solve via a spectral discretization scheme. We demonstrate the accuracy of our method by confirming the quantization of an appropriate interface conductivity modeling transport asymmetry along the interface, and moreover, confirm that this quantity is immune to local perturbations. We also compute the far-field scattering matrix generated by such perturbations and verify that while asymmetric transport is topologically protected the absence of back-scattering is not.

**[02819] Asymmetric transport and topological invariants****Author(s)**:**Guillaume Bal**(University of Chicago)

**Abstract**: Transport asymmetries along interfaces separating insulating bulks have a topological origin. The talk proposes a classification of partial differential systems with topological invariant computed explicitly by a Fedosov-Hörmander formula. Asymmetric transport is associated to another topological invariant whose calculations is less direct. A bulk-edge correspondence states that the two invariants in fact agree. The theory is applied to graphene-based topological insulators. Time permitting, the above spectral analysis will be contrasted with a temporal picture.

**[02822] Quantum dynamics simulation and its application to Hamiltonian learning****Author(s)**:**Di Fang**(Duke University)

**Abstract**: Recent years have witnessed tremendous progress in developing and analyzing quantum algorithms for quantum dynamics simulation (Hamiltonian simulation). The accuracy of quantum dynamics simulation is usually measured by the error of the unitary evolution operator in the operator norm, which in turn depends on the operator norm of the Hamiltonian. However, the operator norm measures the worst-case scenario, while practical simulation concerns the error with respect to a given initial vector or given observables at hand. In this talk, we will discuss a few ways to weaken the strong operator norm dependence in quantum simulation tasks by taking into account the the initial condition and observables. We then discuss how such analysis can be applied in the setting of Hamiltonian learning. Using a Hamiltonian reshaping technique, we propose a first learning algorithm to achieve the Heisenberg limit for efficiently learning an interacting N-qubit local Hamiltonian.