A posteriori error estimates for the Lindblad master equation

• URL: http://arxiv.org/abs/2501.09607v3
• Date: Sun, 02 Feb 2025 13:02:58 GMT
• Title: A posteriori error estimates for the Lindblad master equation
• Authors: Paul-Louis Etienney, Rémi Robin, Pierre Rouchon,
• Abstract summary: We are interested in the simulation of open quantum systems governed by the Lindblad master equation in an infinite-dimensional Hilbert space. Standard approach involves two sequential approximations to derive a differential equation in a finite-dimensional subspace. In this paper, we establish bounds for these two approximations that can be explicitely computed to guarantee the accuracy of the numerical results.
• Score: 0.0
• License:
• Abstract: We are interested in the simulation of open quantum systems governed by the Lindblad master equation in an infinite-dimensional Hilbert space. To simulate the solution of this equation, the standard approach involves two sequential approximations: first, we truncate the Hilbert space to derive a differential equation in a finite-dimensional subspace. Then, we use discrete time-step to obtain a numerical solution to the finite-dimensional evolution. In this paper, we establish bounds for these two approximations that can be explicitely computed to guarantee the accuracy of the numerical results. Through numerical examples, we demonstrate the efficiency of our method, empirically highlighting the tightness of the upper bound. While adaptive time-stepping is already a common practice in the time discretization of the Lindblad equation, we extend this approach by showing how to dynamically adjust the truncation of the Hilbert space. This enables fully adaptive simulations of the density matrix. For large-scale simulations, this approach significantly reduces computational time and relieves users of the challenge of selecting an appropriate truncation.

Related papers

• A data-driven learned discretization approach in finite volume schemes for hyperbolic conservation laws and varying boundary conditions [1.4999444543328293]
  We present a data-driven finite volume method for solving 1D and 2D hyperbolic partial differential equations. New ingredients guarantee computational stability, preserve the accuracy of fine-grid solutions, and enhance overall performance.
  arXiv  Detail & Related papers  (2024-12-10T14:18:30Z)
• Trust-Region Sequential Quadratic Programming for Stochastic Optimization with Random Models [57.52124921268249]
  We propose a Trust Sequential Quadratic Programming method to find both first and second-order stationary points. To converge to first-order stationary points, our method computes a gradient step in each iteration defined by minimizing a approximation of the objective subject. To converge to second-order stationary points, our method additionally computes an eigen step to explore the negative curvature the reduced Hessian matrix.
  arXiv  Detail & Related papers  (2024-09-24T04:39:47Z)
• Full- and low-rank exponential Euler integrators for the Lindblad equation [2.5676905118007407]
  The Lindblad equation is a widely used quantum master equation to model the dynamical evolution of open quantum systems. Full-rank exponential Euler and low-rank exponential Euler are developed for approximating the Lindblad equation that preserve positivity and trace unconditionally.
  arXiv  Detail & Related papers  (2024-08-24T15:11:28Z)
• Solving quantum impurity problems on the L-shaped Kadanoff-Baym contour [0.0]
  We extend the recently developed Grassmann time-evolving matrix product operator (GTEMPO) method to solve quantum impurity problems directly on the Kadanoff-Baym contour. The accuracy of this method is numerically demonstrated against exact solutions in the noninteracting case, and against existing calculations on the real- and imaginary-time axes.
  arXiv  Detail & Related papers  (2024-04-08T11:21:06Z)
• A conservative hybrid physics-informed neural network method for Maxwell-Amp\`{e}re-Nernst-Planck equations [22.81295238376119]
  The proposed hybrid algorithm provides an automated means to determine a proper approximation for dummy variables. The original method is validated for 2-dimensional problems. The proposed method can be readily generalised to cases with one spatial dimension.
  arXiv  Detail & Related papers  (2023-12-10T13:58:41Z)
• Simulating Open Quantum Systems Using Hamiltonian Simulations [4.328210085579236]
  We present a novel method to simulate the Lindblad equation, drawing on the relationship between Lindblad dynamics, differential equations, and Hamiltonian simulations. We derive a sequence of unitary dynamics in an enlarged Hilbert space that can approximate the Lindblad dynamics up to an arbitrarily high order. This unitary representation can then be simulated using a quantum circuit that involves only Hamiltonian simulation and tracing out the ancilla qubits.
  arXiv  Detail & Related papers  (2023-11-27T04:38:13Z)
• Stochastic Marginal Likelihood Gradients using Neural Tangent Kernels [78.6096486885658]
  We introduce lower bounds to the linearized Laplace approximation of the marginal likelihood. These bounds are amenable togradient-based optimization and allow to trade off estimation accuracy against computational complexity.
  arXiv  Detail & Related papers  (2023-06-06T19:02:57Z)
• On optimization of coherent and incoherent controls for two-level quantum systems [77.34726150561087]
  This article considers some control problems for closed and open two-level quantum systems. The closed system's dynamics is governed by the Schr"odinger equation with coherent control. The open system's dynamics is governed by the Gorini-Kossakowski-Sudarshan-Lindblad master equation.
  arXiv  Detail & Related papers  (2022-05-05T09:08:03Z)
• Mean-Square Analysis with An Application to Optimal Dimension Dependence of Langevin Monte Carlo [60.785586069299356]
  This work provides a general framework for the non-asymotic analysis of sampling error in 2-Wasserstein distance. Our theoretical analysis is further validated by numerical experiments.
  arXiv  Detail & Related papers  (2021-09-08T18:00:05Z)
• Lifting the Convex Conjugate in Lagrangian Relaxations: A Tractable Approach for Continuous Markov Random Fields [53.31927549039624]
  We show that a piecewise discretization preserves better contrast from existing discretization problems. We apply this theory to the problem of matching two images.
  arXiv  Detail & Related papers  (2021-07-13T12:31:06Z)
• Optimal oracle inequalities for solving projected fixed-point equations [53.31620399640334]
  We study methods that use a collection of random observations to compute approximate solutions by searching over a known low-dimensional subspace of the Hilbert space. We show how our results precisely characterize the error of a class of temporal difference learning methods for the policy evaluation problem with linear function approximation.
  arXiv  Detail & Related papers  (2020-12-09T20:19:32Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.