arXiv subjects

Xiantao Li

Publications and source records attributed to Xiantao Li.

At least 55 records · Page 3Linked to original sources

arXiv2023

Succinct Description and Efficient Simulation of Non-Markovian Open Quantum Systems

Non-Markovian open quantum systems represent the most general dynamics when the quantum system is coupled with a bath environment. The quantum dynamics arising from many important applications are non-Markovian. Although for special cases, such as Hamiltonian evolution and Lindblad evolution, quantum simulation algorithms have been extensively studied, efficient quantum simulations for the dynamics of non-Markovian open quantum systems remain underexplored. The most immediate obstacle for studying such systems is the lack of a universal succinct description of their dynamics. In this work, we fulfill the gap of studying such dynamics by 1) providing a succinct representation of the dynamics of non-Markovian open quantum systems with quantifiable error, and 2) developing an efficient quantum algorithm for simulating such dynamics with cost $O(t\, \mathrm{polylog}(t/ε))$ for evolution time $t$ and precision $ε$. Our derivation of the succinct representation is based on stochastic Schrödinger equations, which could lead to new alternatives to deal with open quantum systems as well.

quant-ph↗

arXiv2022

Stochastic Algorithms for Self-consistent Calculations of Electronic Structures

The convergence property of a stochastic algorithm for the self-consistent field (SCF) calculations of electron structures is studied. The algorithm is formulated by rewriting the electron charges as a trace/diagonal of a matrix function, which is subsequently expressed as a statistical average. The function is further approximated by using a Krylov subspace approximation. As a result, each SCF iteration only samples one random vector without having to compute all the orbitals. We consider the common practice of SCF iterations with damping and mixing. We prove with appropriate assumptions that the iterations converge in the mean-square sense, when the stochastic error has an almost sure bound. We also consider the scenario when such an assumption is weakened to a second moment condition, and prove the convergence in probability.

math.NA↗

arXiv2022

Enabling Quantum Speedup of Markov Chains using a Multi-level Approach

Quantum speedup for mixing a Markov chain can be achieved based on the construction of slowly-varying $r$ Markov chains where the initial chain can be easily prepared and the spectral gaps have uniform lower bound. The overall complexity is proportional to $r$. We present a multi-level approach to construct such a sequence of $r$ Markov chains by varying a resolution parameter $h.$ We show that the density function of a low-resolution Markov chain can be used to warm start the Markov chain with high resolution. We prove that in terms of the chain length the new algorithm has $O(1)$ complexity rather than $O(r).$

quant-ph↗

arXiv2022

Some Error Analysis for the Quantum Phase Estimation Algorithms

This paper is concerned with the phase estimation algorithm in quantum computing algorithms, especially the scenarios where (1) the input vector is not an eigenvector; (2) the unitary operator is not exactly implemented; (3) random approximations are used for the unitary operator, e.g., the QDRIFT method. We characterize the probability of computing the phase values in terms of the consistency error, including the residual error, Trotter splitting error, or statistical mean-square error. In the first two cases, we show that in order to obtain the phase value with {error less or equal to $2^{-n}$ } and probability at least $1-ε$, the required number of qubits is $ t \geq n + \log \big(2 + \frac{δ^2 }{2 εΔ\!E^2 } \big).$ The parameter $δ$ quantifies the error associated with the inexact eigenvector and/or the unitary operator, and $Δ\! E$ characterizes the spectral gap, i.e., the separation from the rest of the phase values. For the third case, we found a similar estimate, but the number of random steps has to be sufficiently large.

quant-ph↗

arXiv2022

On quantum algorithms for the Schrödinger equation in the semi-classical regime

Solving the time-dependent Schrödinger equation is an important application area for quantum algorithms. We consider Schrödinger's equation in the semi-classical regime. Here the solutions exhibit strong multiple-scale behavior due to a small parameter $\hbar$, in the sense that the dynamics of the quantum states and the induced observables can occur on different spatial and temporal scales. Such a Schrödinger equation finds many applications, including in Born-Oppenheimer molecular dynamics and Ehrenfest dynamics. This paper considers quantum analogues of pseudo-spectral (PS) methods on classical computers. Estimates on the gate counts in terms of $\hbar$ and the precision $\varepsilon$ are obtained. It is found that the number of required qubits, $m$, scales only logarithmically with respect to $\hbar$. When the solution has bounded derivatives up to order $\ell$, the symmetric Trotting method has gate complexity $\mathcal{O}\Big({ (\varepsilon \hbar)^{-\frac12} \mathrm{polylog}(\varepsilon^{-\frac{3}{2\ell}} \hbar^{-1-\frac{1}{2\ell}})}\Big),$ provided that the diagonal unitary operators in the pseudo-spectral methods can be implemented with $\mathrm{poly}(m)$ operations. When physical observables are the desired outcomes, however, the step size in the time integration can be chosen independently of $\hbar$. The gate complexity in this case is reduced to $\mathcal{O}\Big({\varepsilon^{-\frac12} \mathrm{polylog}( \varepsilon^{-\frac3{2\ell}} \hbar^{-1} )}\Big),$ with $\ell$ again indicating the smoothness of the solution.

quant-ph↗

arXiv2022

A Local Convergence Theory for the Stochastic Gradient Descent Method in Non-Convex Optimization With Non-isolated Local Minima

Loss functions with non-isolated minima have emerged in several machine learning problems, creating a gap between theory and practice. In this paper, we formulate a new type of local convexity condition that is suitable to describe the behavior of loss functions near non-isolated minima. We show that such condition is general enough to encompass many existing conditions. In addition we study the local convergence of the SGD under this mild condition by adopting the notion of stochastic stability. The corresponding concentration inequalities from the convergence analysis help to interpret the empirical observation from some practical training results.

cs.LG↗

arXiv2022

Stability Preserving Data-driven Models With Latent Dynamics

In this paper, we introduce a data-driven modeling approach for dynamics problems with latent variables. The state-space of the proposed model includes artificial latent variables, in addition to observed variables that can be fitted to a given data set. We present a model framework where the stability of the coupled dynamics can be easily enforced. The model is implemented by recurrent cells and trained using backpropagation through time. Numerical examples using benchmark tests from order reduction problems demonstrate the stability of the model and the efficiency of the recurrent cell implementation. As applications, two fluid-structure interaction problems are considered to illustrate the accuracy and predictive capability of the model.

math.OC↗

arXiv2021

Projection based model reduction for the immersed boundary method

Fluid-structure interactions are central to many bio-molecular processes, and they impose a great challenge for computational and modeling methods. In this paper, we consider the immersed boundary method (IBM) for biofluid systems, and to alleviate the computational cost, we apply reduced-order techniques to eliminate the degrees of freedom associated with a large number of fluid variables. We show how reduced models can be derived using Petrov-Galerkin projection and subspaces that maintain the incompressibility condition. More importantly, the reduced-order model is shown to preserve the Lyapunov stability. We also address the practical issue of computing coefficient matrices in the reduced-order model using an interpolation technique. The efficiency and robustness of the proposed formulation are examined with test examples from various applications.

math.NA↗

arXiv2021

A Partially Random Trotter Algorithm for Quantum Hamiltonian Simulations

Given the Hamiltonian, the evaluation of unitary operators has been at the heart of many quantum algorithms. Motivated by existing deterministic and random methods, we present a hybrid approach, where Hamiltonians with large amplitude are evaluated at each time step, while the remaining terms are evaluated at random. The bound for the mean square error is obtained, together with a concentration bound. The mean square error consists of a variance term and a bias term, arising respectively from the random sampling of the Hamiltonian terms and the operator splitting error. Leveraging on the bias/variance trade-off, the error can be minimized by balancing the two. The concentration bound provides an estimate on the number of gates. The estimates are verified by using numerical experiments on classical computers.

quant-ph↗

arXiv2021

A Machine-Learning Method for Time-Dependent Wave Equations over Unbounded Domains

Time-dependent wave equations represent an important class of partial differential equations (PDE) for describing wave propagation phenomena, which are often formulated over unbounded domains. Given a compactly supported initial condition, classical numerical methods reduce such problems to bounded domains using artificial boundary condition (ABC). In this work, we present a machine-learning method to solve this type of equations as an alternative to ABCs. Specifically, the mapping from the initial conditions to the PDE solution is represented by a neural network, trained using wave packets that are parameterized by their band width and wave numbers. The accuracy is tested for both the second-order wave equation and the Schrodinger equation, including the nonlinear Schrodinger equation. We examine the accuracy from both interpolations and extrapolations. For initial conditions lying in the training set, the learned map has good interpolation accuracy, due to the approximation property of deep neural networks. The learned map also exhibits some good extrapolation accuracy. We also demonstrate the effectiveness of the method for problems in irregular domains. Overall, the proposed method provides an interesting alternative for finite-time simulation of wave propagation.

math.NA↗

arXiv2021

A Projection-based Reduced-order Method for Electron Transport Problems with Long-range Interactions

Long-range interactions play a central role in electron transport. At the same time, they present a challenge for direct computer simulations, since sufficiently large portions of the bath have to be included in the computation to accurately compute the Coulomb potential. This article presents a reduced-order approach, by deriving an open quantum model for the reduced density-matrix. To treat the transient dynamics, the problem is placed in a reduced-order framework. The dynamics, described by the Liouville von Neumann equation, is projected to subspaces using a Petrov-Galerkin projection. In order to recover the global electron density profile as a vehicle to compute the Coulomb potential, we propose a domain decomposition approach, where the computational domain also includes segments of the bath that are selected using logarithmic grids. This approach leads to a multi-component self-energy that enters the effective Hamiltonian. We demonstrate the accuracy of the reduced model using a molecular junction built from a Lithium chains.

physics.comp-ph↗

arXiv2021

Error Bounds of the Invariant Statistics in Machine Learning of Ergodic Itô Diffusions

This paper studies the theoretical underpinnings of machine learning of ergodic Itô diffusions. The objective is to understand the convergence properties of the invariant statistics when the underlying system of stochastic differential equations (SDEs) is empirically estimated with a supervised regression framework. Using the perturbation theory of ergodic Markov chains and the linear response theory, we deduce a linear dependence of the errors of one-point and two-point invariant statistics on the error in the learning of the drift and diffusion coefficients. More importantly, our study shows that the usual $L^2$-norm characterization of the learning generalization error is insufficient for achieving this linear dependence result. We find that sufficient conditions for such a linear dependence result are through learning algorithms that produce a uniformly Lipschitz and consistent estimator in the hypothesis space that retains certain characteristics of the drift coefficients, such as the usual linear growth condition that guarantees the existence of solutions of the underlying SDEs. We examine these conditions on two well-understood learning algorithms: the kernel-based spectral regression method and the shallow random neural networks with the ReLU activation function.

cs.LG↗

arXiv2020

Estimating linear response statistics using orthogonal polynomials: An RKHS formulation

We study the problem of estimating linear response statistics under external perturbations using time series of unperturbed dynamics. Based on the fluctuation-dissipation theory, this problem is reformulated as an unsupervised learning task of estimating a density function. We consider a nonparametric density estimator formulated by the kernel embedding of distributions with "Mercer-type" kernels, constructed based on the classical orthogonal polynomials defined on non-compact domains. While the resulting representation is analogous to Polynomial Chaos Expansion (PCE), the connection to the reproducing kernel Hilbert space (RKHS) theory allows one to establish the uniform convergence of the estimator and to systematically address a practical question of identifying the PCE basis for a consistent estimation. We also provide practical conditions for the well-posedness of not only the estimator but also of the underlying response statistics. Finally, we provide a statistical error bound for the density estimation that accounts for the Monte-Carlo averaging over non-i.i.d time series and the biases due to a finite basis truncation. This error bound provides a means to understand the feasibility as well as limitation of the kernel embedding with Mercer-type kernels. Numerically, we verify the effectiveness of the estimator on two stochastic dynamics with known, yet, non-trivial equilibrium densities.

math.ST↗

arXiv2020

Linear Response Based Parameter Estimation in the Presence of Model Error

Recently, we proposed a method to estimate parameters of stochastic dynamics based on the linear response statistics. The method rests upon a nonlinear least-squares problem that takes into account the response properties that stem from the Fluctuation-Dissipation Theory. In this article, we address an important issue that arises in the presence of model error. In particular, when the equilibrium density function is high dimensional and non-Gaussian, and in some cases, is unknown, the linear response statistics are inaccessible. We show that this issue can be resolved by fitting the imperfect model to appropriate marginal linear response statistics that can be approximated using the available data and parametric or nonparametric models. The effectiveness of the parameter estimation approach is demonstrated in the context of molecular dynamical models (Langevin dynamics) with a non-uniform temperature profile, where the modeling error is due to coarse-graining, and a PDE (non-Langevin dynamics) that exhibits spatiotemporal chaos, where the model error arises from a severe spectral truncation. In these examples, we show how the imperfect models, the Langevin equation with parameters estimated using the proposed scheme, can predict the nonlinear response statistics of the underlying dynamics under admissible external disturbances.

math.NA↗

arXiv2020

Random Batch Algorithms for Quantum Monte Carlo simulations

Random batch algorithms are constructed for quantum Monte Carlo simulations. The main objective is to alleviate the computational cost associated with the calculations of two-body interactions, including the pairwise interactions in the potential energy, and the two-body terms in the Jastrow factor. In the framework of variational Monte Carlo methods, the random batch algorithm is constructed based on the over-damped Langevin dynamics, so that updating the position of each particle in an $N$-particle system only requires $\mathcal{O}(1)$ operations, thus for each time step the computational cost for $N$ particles is reduced from $\mathcal{O}(N^2)$ to $\mathcal{O}(N)$. For diffusion Monte Carlo methods, the random batch algorithm uses an energy decomposition to avoid the computation of the total energy in the branching step. The effectiveness of the random batch method is demonstrated using a system of liquid ${}^4$He atoms interacting with a graphite surface.

physics.comp-ph↗

arXiv2020

The strong convergence of operator-splitting methods for the Langevin dynamics model

We study the strong convergence of some operator-splitting methods for the Langevin dynamics model with additive noise. It will be shown that a direct splitting of deterministic and random terms, including the symmetric splitting methods, only offers strong convergence of order 1. To improve the order of strong convergence, a new class of operator-splitting methods based on Kunita's solution representation are proposed. We present stochastic algorithms with strong orders up to 3. Both mathematical analysis and numerical evidence are provided to verify the desired order of accuracy.

math.NA↗

arXiv2020

Data-driven molecular modeling with the generalized Langevin equation

The complexity of molecular dynamics simulations necessitates dimension reduction and coarse-graining techniques to enable tractable computation. The generalized Langevin equation (GLE) describes coarse-grained dynamics in reduced dimensions. In spite of playing a crucial role in non-equilibrium dynamics, the memory kernel of the GLE is often ignored because it is difficult to characterize and expensive to solve. To address these issues, we construct a data-driven rational approximation to the GLE. Building upon previous work leveraging the GLE to simulate simple systems, we extend these results to more complex molecules, whose many degrees of freedom and complicated dynamics require approximation methods. We demonstrate the effectiveness of our approximation by testing it against exact methods and comparing observables such as autocorrelation and transition rates.

physics.comp-ph↗

arXiv2020

Markovian Embedding Procedures for Non-Markovian Stochastic Schrödinger Equations

We present embedding procedures for the non-Markovian stochastic Schrödinger equations, arising from studies of quantum systems coupled with bath environments. By introducing auxiliary wave functions, it is demonstrated that the non-Markovian dynamics can be embedded in extended, but Markovian, stochastic models. Two embedding procedures are presented. The first method leads to nonlinear stochastic equations, the implementation of which is much more efficient than the non-Markovian stochastic Schrödinger equations. The stochastic Schrödinger equations obtained from the second procedure involve more auxiliary wave functions, but the equations are linear, and we derive the corresponding generalized quantum master equation for the density-matrix. The accuracy of the embedded models is ensured by fitting to the power spectrum. The stochastic force is represented using a linear superposition of Ornstein-Uhlenbeck processes, which are incorporated as multiplicative noise in the auxiliary Schrödinger equations. The asymptotic behavior of the spectral density in the low frequency regime is preserved by using correlated stochastic processes. The approximations are verified by using a spin-boson system as a test example.

physics.comp-ph↗