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Coupled Lindblad pseudomode theory for simulating open quantum systems

Zhen Huang, Gunhee Park, Garnet Kin-Lic Chan, Lin Lin

Abstract

Coupled Lindblad pseudomode theory is a promising approach for simulating non-Markovian quantum dynamics on both classical and quantum platforms, with dynamics that can be realized as a quantum channel. We provide theoretical evidence that the number of coupled pseudomodes only needs to scale as $\mathrm{polylog}(T/\varepsilon)$ in the simulation time $T$ and precision $\varepsilon$. Inspired by the realization problem in control theory, we also develop a robust numerical algorithm for constructing the coupled modes that avoids the non-convex optimization required by existing approaches. We demonstrate the effectiveness of our method by computing population dynamics and absorption spectra for the spin-boson model. This work provides a significant theoretical and computational improvement to the coupled Lindblad framework, which impacts a broad range of applications from classical simulations of quantum impurity problems to quantum simulations on near-term quantum platforms.

Coupled Lindblad pseudomode theory for simulating open quantum systems

Abstract

Coupled Lindblad pseudomode theory is a promising approach for simulating non-Markovian quantum dynamics on both classical and quantum platforms, with dynamics that can be realized as a quantum channel. We provide theoretical evidence that the number of coupled pseudomodes only needs to scale as in the simulation time and precision . Inspired by the realization problem in control theory, we also develop a robust numerical algorithm for constructing the coupled modes that avoids the non-convex optimization required by existing approaches. We demonstrate the effectiveness of our method by computing population dynamics and absorption spectra for the spin-boson model. This work provides a significant theoretical and computational improvement to the coupled Lindblad framework, which impacts a broad range of applications from classical simulations of quantum impurity problems to quantum simulations on near-term quantum platforms.

Paper Structure

This paper contains 7 sections, 1 theorem, 18 equations, 7 figures, 1 table.

Table of Contents

  1. Realization-based coupled Lindblad modes construction
  2. Coupled Lindblad pseudomodes theory for multi-site cases
  3. Coupled Lindblad pseudomode theory for fermionic systems
  4. TD-DMRG based simulation
  5. Numerical experiments on the fermionic Anderson impurity model
  6. Additional information on benchmarking BCF fitting
  7. Additional Information on calculating the absorption spectrum

Key Result

Theorem 1

Let $\hat{\rho}_{\text{S}}^{\text{c}}(t)$ and $\hat{\rho}_{\text{S}}^{\text{q}}(t)$ denote the reduced system density operators obtained from the coupled Lindblad and quasi-Lindblad theory, respectively. If the BCF coincide, then the reduced dynamics are identical: Furthermore, if the following feasibility condition holds, then there exists a coupled Lindblad BCF $C^{\text{c}}(t)$, with the same

Figures (7)

  • Figure 1: (a) For a fixed precision $\varepsilon = 10^{-6}$ in fitting $C(t)$ for $t \in [0, T]$, we plot the number of modes required, $N$, against the maximum simulation time $T$. The number of coupled Lindblad and quasi-Lindblad pseudomodes scales as $\mathcal{O}(\log T)$ in contrast to the $\mathcal{O}(T)$ scaling in the unitary and Lorentzian modes. (b) For a fixed $T = 10$, we plot $\varepsilon$ versus $N$, where the coupled Lindblad and quasi-Lindblad methods achieve a significantly faster convergence rate.
  • Figure 2: (a) Population $n_0(t)$ and its relative error for the spin-boson model dynamics. (b) Spectral density $J(\omega)$, and its fitting using coupled modes $N=4$ and $N=10$. Both plots are compared with results for $N=10$ extracted from Ref. LednevGarciaFeist2024.
  • Figure 3: Normalized absorption spectrum $S(\omega)$ for the dimer model with two different environments $J_0(\omega)$ (a, c) and $J_1(\omega)$ (b, d), at zero (a, b) and finite temperature (77K, (c, d)).
  • Figure S1: Numerical experiments on the Fermionic Anderson impurity model with semicircular bath spectral density using the coupled Lindblad approach. (Left) Results of BCF fitting. (Right) Dynamics of $n_\uparrow(t)$.
  • Figure S2: Comparison of unitary, Lorentzian, coupled Lindblad (this work) and quasi-Lindblad pseudomodes for fitting the same BCF using $N=4$ modes.
  • ...and 2 more figures

Theorems & Definitions (2)

  • Theorem 1
  • proof