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NIPG-DG schemes for transformed master equations modeling open quantum systems

Jose A. Morales Escalante

TL;DR

This paper develops a discontinuous Galerkin solver for a transformed master equation obtained by Fourier transforming a Lindblad/Wigner-Fokker-Planck model of open quantum systems. By converting the nonlocal pseudo-differential potential term into a local product, the method treats the problem as a convective-diffusive system in transformed coordinates and solves it in a DG framework with real/imaginary density matrix components. Convergence and accuracy are demonstrated for a harmonic benchmark with analytic steady state, and the approach is extended to nonharmonic linear and quartic potentials, showing robust performance and reduced computational cost compared to the WFP formulation. The results have implications for efficient simulations of continuous-variable open quantum systems and may inform future work in higher dimensions, uncertainty quantification, and NISQ-era applications.

Abstract

This work presents a numerical analysis of a Discontinuous Galerkin (DG) method for a transformed master equation modeling an open quantum system: a quantum sub-system interacting with a noisy environment. It is shown that the presented transformed master equation has a reduced computational cost in comparison to a Wigner-Fokker-Planck model of the same system for the general case of non-harmonic potentials via DG schemes. Specifics of a Discontinuous Galerkin (DG) numerical scheme adequate for the system of convection-diffusion equations obtained for our Lindblad master equation in position basis are presented. This lets us solve computationally the transformed system of interest modeling our open quantum system problem. The benchmark case of a harmonic potential is then presented, for which the numerical results are compared against the analytical steady-state solution of this problem. Two non-harmonic cases are then presented: the linear and quartic potentials are modeled via our DG framework, for which we show our numerical results.

NIPG-DG schemes for transformed master equations modeling open quantum systems

TL;DR

This paper develops a discontinuous Galerkin solver for a transformed master equation obtained by Fourier transforming a Lindblad/Wigner-Fokker-Planck model of open quantum systems. By converting the nonlocal pseudo-differential potential term into a local product, the method treats the problem as a convective-diffusive system in transformed coordinates and solves it in a DG framework with real/imaginary density matrix components. Convergence and accuracy are demonstrated for a harmonic benchmark with analytic steady state, and the approach is extended to nonharmonic linear and quartic potentials, showing robust performance and reduced computational cost compared to the WFP formulation. The results have implications for efficient simulations of continuous-variable open quantum systems and may inform future work in higher dimensions, uncertainty quantification, and NISQ-era applications.

Abstract

This work presents a numerical analysis of a Discontinuous Galerkin (DG) method for a transformed master equation modeling an open quantum system: a quantum sub-system interacting with a noisy environment. It is shown that the presented transformed master equation has a reduced computational cost in comparison to a Wigner-Fokker-Planck model of the same system for the general case of non-harmonic potentials via DG schemes. Specifics of a Discontinuous Galerkin (DG) numerical scheme adequate for the system of convection-diffusion equations obtained for our Lindblad master equation in position basis are presented. This lets us solve computationally the transformed system of interest modeling our open quantum system problem. The benchmark case of a harmonic potential is then presented, for which the numerical results are compared against the analytical steady-state solution of this problem. Two non-harmonic cases are then presented: the linear and quartic potentials are modeled via our DG framework, for which we show our numerical results.
Paper Structure (14 sections, 68 equations, 13 figures, 10 tables)

This paper contains 14 sections, 68 equations, 13 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Math Model: Transformed Master Equation for open quantum systems
  3. Methodology: DG Formulation for Master Equations in transformed position coordinates
  4. DG method for a transformed Master Equation
  5. Computational Cost of Master Equations vs WFP via DG methods
  6. Numerical Results
  7. Boundary Conditions (BC) for the Numerical Problem
  8. Benchmark Problem: Harmonic Potential with Steady State BC
  9. Convergence and Error Analysis for NIPG solutions at $t=2.0$
  10. Convergence and Error Analysis for NIPG solutions at $t=50.0$
  11. Comparing Homogeneous vs. Steady-state BC for Benchmark Harmonic Case
  12. Linear Potential
  13. Quartic Potential
  14. Conclusions

Figures (13)

  • Figure 1: Plot of the real component of the density matrix (in convenient position coordinates) of the harmonic oscillator groundstate (the imaginary component is zero and therefore omitted).
  • Figure 2: Plot of the real component of the density matrix (in convenient position coordinates) of the steady state for our transformed master equation under a harmonic potential.
  • Figure 3: Initial condition for the real (left) and imaginary (right) parts of the density matrix in convenient coordinates, corresponding to a harmonic ground state (projected into the $V_h^1$ DG FE space). Remark: Color scale differs between right picture and left one.
  • Figure 4: Numerical solution of the real (left) and imaginary (right) parts of the density matrix in convenient coordinates under a harmonic potential after an evolution time of $t=2.0$, solved by DG. Remark: Color scale differs between right picture and left one.
  • Figure 5: Numerical solution of the real (left) and imaginary (right) parts of the density matrix in convenient coordinates under a harmonic potential after an evolution time of $t=10.0$, solved by DG. Remark: Color scale differs between right picture and left one.
  • ...and 8 more figures