Full- and low-rank exponential Euler integrators for the Lindblad equation

TL;DR

This work introduces two exponential Euler integrators for the Lindblad equation: a full-rank method (FREE) that unconditionally preserves positivity and trace by solving a Lyapunov-type equation at each step, and a low-rank method (LREE) that compresses the density matrix via factorization $\varrho_n = Z_n Z_n^{\dagger}$ with rank truncation. The authors prove sharp first-order accuracy in time and establish trace-preservation and positivity properties, with a clear analysis of the low-rank perturbations. Numerical experiments on Ising-type Hamiltonians and GHZ initial states demonstrate both accuracy and significant speedups of LREE for large-scale problems, and comparisons with QuTip reveal practical advantages in positivity guarantees and memory usage. The results offer scalable, physically consistent solvers for open quantum systems, particularly beneficial when the Hilbert-space dimension is large.

Abstract

The Lindblad equation is a widely used quantum master equation to model the dynamical evolution of open quantum systems whose states are described by density matrices. These solution matrices are characterized by semi-positiveness and trace preserving properties, which must be guaranteed in any physically meaningful numerical simulation. In this paper, novel full- and low-rank exponential Euler integrators are developed for approximating the Lindblad equation that preserve positivity and trace unconditionally. Theoretical results are presented that provide sharp error estimates for the two classes of exponential integration methods. Results of numerical experiments are discussed that illustrate the effectiveness of the proposed schemes, beyond present state-of-the-art capabilities.

Figures (11)

• Figure 6.1: Numerical results of the FREE integrator for the Lindblad equation with Hamiltonian \ref{['equ6.1']} ($d=4$, $K=4$, $a=1.5$, $b=0.5$, $\gamma_k=\gamma$, $g_{kl}(t)\equiv g$). Top left: relative errors at $T=1$ vs step sizes for different $g$ and $\gamma$. Top right: evolutions of $\hbox{Tr}(\rho_n)-1$ vs time for different $\tau$ with fixed $g=1$, $\gamma=0.01$ and $T=20$. Bottom left: evolutions of the populations $\rho_{1,1}$ and $\rho_{256,256}$ with $\tau=0.1$, $g=1$, $\gamma=0.01$ and $T=20$. Bottom right: evolutions of the populations $\rho_{3,3}$ and $\rho_{8,8}$ with $\tau=0.1$, $g=1$, $\gamma=0.01$ and $T=20$.
• Figure 6.2: Numerical results of the FREE integrator for the Lindblad equation with time-dependent Hamiltonian \ref{['equ6.1']} ($d=6$, $K=3$, $a=1$, $b=1$, $\gamma_k=\gamma$, $g_{kl}(t)= \delta_{k,l-1}\cdot g(t)$). Top left: relative errors at $T=1$ vs step sizes for different $g(t)$ and $\gamma$. Top right: evolutions of $\hbox{Tr}(\rho_n)-1$ vs time for different $\tau$ with $g(t)=(1+t)^{\frac{1}{4}}$, $\gamma=0.05$ and $T=20$. Bottom left: evolutions of the populations $\rho_{1,1}$ and $\rho_{216,216}$ with $\tau=0.1$, $g(t)=(1+t)^{\frac{1}{4}}$, $\gamma=0.05$ and $T=20$. Bottom right: evolutions of the populations $\rho_{4,4}$ and $\rho_{11,11}$ with $\tau=0.1$, $g(t)=(1+t)^{\frac{1}{4}}$, $\gamma=0.05$ and $T=20$.
• Figure 6.3: Numerical results of the LREE scheme with $tol_1=10^{-10}$ and $tol_2=10^{-10}$ for the Lindblad equation with Hamiltonian \ref{['equ6.1']}. Left: relative errors at $T=1$ vs step sizes for the Lindblad problem with $d=4$, $K=4$, $a=1.5$, $b=0.5$, $\gamma_k=\gamma=0.01$, $g_{kl}(t)\equiv g=1$. Right: relative errors at $T=1$ vs step sizes for the Lindblad problem with $d=6$, $K=3$, $a=1$, $b=1$, $\gamma_k=\gamma=0.05$, $g_{kl}(t)= \delta_{k,l-1}\cdot(1+t)^{\frac{1}{4}}$.
• Figure 6.4: Numerical results of the LREE scheme with $\delta=0$ and $tol_1=10^{-10}$ for the Lindblad equation with Hamiltonian \ref{['equ6.1']}. Left: relative errors at $T=1$ vs step sizes for the Lindblad problem with $d=4$, $K=4$, $a=1.5$, $b=0.5$, $\gamma_k=\gamma=0.01$, $g_{kl}(t)\equiv g=1$. Right: relative errors at $T=1$ vs step sizes for the Lindblad problem with $d=6$, $K=3$, $a=1$, $b=1$, $\gamma_k=\gamma=0.05$, $g_{kl}(t)= \delta_{k,l-1}\cdot(1+t)^{\frac{1}{4}}$.
• Figure 6.5: Numerical results of the LREE scheme with $\delta=0$ and $tol_2=10^{-10}$ for the Lindblad equation with Hamiltonian \ref{['equ6.1']}. Left: relative errors at $T=1$ vs step sizes for the Lindblad problem with $d=4$, $K=4$, $a=1.5$, $b=0.5$, $\gamma_k=\gamma=0.01$, $g_{kl}(t)\equiv g=1$. Right: relative errors at $T=1$ vs step sizes for the Lindblad problem with $d=6$, $K=3$, $a=1$, $b=1$, $\gamma_k=\gamma=0.05$, $g_{kl}(t)= \delta_{k,l-1}\cdot(1+t)^{\frac{1}{4}}$.

Theorems & Definitions (22)

• Remark 2.1
• Remark 2.2
• Theorem 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.3
• proof
• Remark 3.4
• Lemma 4.1