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Unified error bounds for perturbations of non-Markovian open quantum systems in Gaussian environments

Zhen Huang, Yuanran Zhu, Gunhee Park, Lin Lin

TL;DR

This work addresses the challenge of controlling perturbative errors in non-Markovian open quantum system dynamics for Gaussian environments, providing a rigorous, unified superoperator framework that handles unitary, Lindblad, and quasi-Lindblad evolutions across bosonic and fermionic baths. It derives a Grönwall-type bound and a substantially sharper improved bound for reduced-system observables under bath-correlator perturbations, with the key quantities expressed in terms of two-point bath correlation functions $C_{\alpha,\alpha'}(t-t')$ and their perturbations. The authors prove the main result rigorously and demonstrate its applicability to spin-boson and fermionic impurity models, including growth estimates of environment correlators under infrared regularity and the quasi-Lindblad setting, making it relevant for pseudomode and HEOM approaches. Together, these results enable long-time, controlled simulations of non-Markovian open quantum systems and provide theoretical support for modern auxiliary-dynamics techniques used in quantum technologies.

Abstract

We present perturbative error bounds for the non-Markovian dynamics of observables in open quantum systems interacting with Gaussian environments, governed by a general Liouville dynamics. This extends the work of [Mascherpa et al., Phys. Rev. Lett. 118, 100401, 2017], which demonstrated qualitatively tighter bounds over the standard Grönwall-type inequality for unitary system-bath evolution. Our results apply to systems with both bosonic and fermionic environments. Our approach utilizes a superoperator formalism, which avoids the need for formal coherent state path integral calculations, or the dilation of Lindblad dynamics into an equivalent unitary framework with infinitely many degrees of freedom. This enables a unified treatment of a wide range of open quantum systems. These findings provide a solid theoretical basis for various recently developed pseudomode methods in simulating open quantum system dynamics.

Unified error bounds for perturbations of non-Markovian open quantum systems in Gaussian environments

TL;DR

This work addresses the challenge of controlling perturbative errors in non-Markovian open quantum system dynamics for Gaussian environments, providing a rigorous, unified superoperator framework that handles unitary, Lindblad, and quasi-Lindblad evolutions across bosonic and fermionic baths. It derives a Grönwall-type bound and a substantially sharper improved bound for reduced-system observables under bath-correlator perturbations, with the key quantities expressed in terms of two-point bath correlation functions and their perturbations. The authors prove the main result rigorously and demonstrate its applicability to spin-boson and fermionic impurity models, including growth estimates of environment correlators under infrared regularity and the quasi-Lindblad setting, making it relevant for pseudomode and HEOM approaches. Together, these results enable long-time, controlled simulations of non-Markovian open quantum systems and provide theoretical support for modern auxiliary-dynamics techniques used in quantum technologies.

Abstract

We present perturbative error bounds for the non-Markovian dynamics of observables in open quantum systems interacting with Gaussian environments, governed by a general Liouville dynamics. This extends the work of [Mascherpa et al., Phys. Rev. Lett. 118, 100401, 2017], which demonstrated qualitatively tighter bounds over the standard Grönwall-type inequality for unitary system-bath evolution. Our results apply to systems with both bosonic and fermionic environments. Our approach utilizes a superoperator formalism, which avoids the need for formal coherent state path integral calculations, or the dilation of Lindblad dynamics into an equivalent unitary framework with infinitely many degrees of freedom. This enables a unified treatment of a wide range of open quantum systems. These findings provide a solid theoretical basis for various recently developed pseudomode methods in simulating open quantum system dynamics.

Paper Structure

This paper contains 25 sections, 33 theorems, 254 equations, 1 table.

Table of Contents

  1. Introduction
  2. Error bounds of open quantum systems under Liouville dynamics
  3. Preliminaries
  4. Tracing out the environment
  5. Grönwall-type error bound
  6. Improved error bound
  7. Proof of Theorem \ref{['thm:main_error_bound']}
  8. Applications to spin-boson and fermionic models described by unitary and Lindblad dynamics
  9. Spin-boson model
  10. Bosonic environment
  11. System-environment coupling
  12. Growth estimate of environment correlators
  13. Dynamics and error bound
  14. Fermionic impurity model
  15. Fermionic environment
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $C(t-t')$ and $C'(t-t') = C(t-t')+\Delta C(t-t')$ be two-point BCFs corresponding to two different environments. For a bounded system operator $\hat{O}_{\text{S}}$, let $O_{\text{S}}(t)$, $O_{\text{S}}'(t)$ be the expectation value of system observables corresponding to the two environments. The where $\|\hat{O}_{\text{S}}\|$ is the operator norm of $\hat{O}_{\text{S}}$, $\mathcal{M}_1= \max\

Theorems & Definitions (77)

  • Theorem 1.1: Grönwall-type error bound for system observables
  • Theorem 1.2: Main result, for system observables, also see \ref{['thm:main_error_bound']}
  • Lemma 2.1
  • Definition 2.3: $n$-point environment superoperator correlation functions
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Remark 2.1
  • Remark 2.2
  • ...and 67 more