Well-conditioned iterative methods for large open quantum systems
Abstract
Markovian open quantum systems are well modeled by the Lindblad Master Equation (ME) d/(d t) ρt = ℒ ρt, where ℒ is a linear (super-)operator and ρt is the system state, a positive matrix. When designing or characterizing a quantum system, one is usually interested in the steady state ρ∞ (such that ℒ ρ∞ = 0), the first few excited states, and trajectories t↦ ρt. In finite dimension, ρt is an n× n matrix, ℒ thus typically costs n4 to store explicitly as a dense matrix, and O(n6) to diagonalize or invert exactly, making standard linear algebraic techniques expensive for large systems. However, ℒ usually costs only O(n3) to apply. This makes iterative methods appealing, but they do not work without a good preconditioner. In this article, our main observation is that a part of the Lindblad equation, corresponding to the so-called no-jump evolution 𝒮, can be inverted efficiently. Using this inverse map, we introduce an auxiliary completely positive trace-preserving (CPTP) map Φ whose fixed point is directly related to ρ∞, all the other eigenvalues having smaller magnitude. The map Φ is thus well suited to iterative methods, and ρ∞ can be found in a few Arnoldi iterations. Using the same inverse map 𝒮−1 as preconditioner, we compute the low-lying spectrum efficiently via shift-invert Arnoldi, and, as a proof of concept, build an implicit time integrator that is competitive on stiff systems in the low-precision regime. For the steady-state and low excited states problems, our methods scale like O(n3) per iteration and offer state-of-the-art performance on CPU and GPU.
Related
- An overview of machine learning-enhanced iterative methods for systems of linear and nonlinear equations
- Methods for Deriving Nonlinear Kinetic Equations from the Dynamics of Quantum Systems
- Universal classical and quantum fluctuations in the large deviations of current of noisy quantum systems: Quantum symmetric simple exclusion and inclusion processes
- Grad-div stabilized finite element Picard-Newton iterative methods for the Navier-Stokes equations
- Iterative Methods for Solving New Generalized Absolute Value Equations
BibTeX
@misc{beugnot2026wellconditioned,
title = {{Well-conditioned iterative methods for large open quantum systems}},
author = {Gaspard Beugnot and Paul Gregory and Rémi Robin and Antoine Tilloy},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30860},
eprint = {2608.30860},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30860},
}
RIS
TY - UNPB
TI - Well-conditioned iterative methods for large open quantum systems
AU - Gaspard Beugnot
AU - Paul Gregory
AU - Rémi Robin
AU - Antoine Tilloy
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30860
UR - https://arxiv.org/abs/2608.30860
AB - Markovian open quantum systems are well modeled by the Lindblad Master Equation (ME) d/(d t) ρ_t = ℒ ρ_t, where ℒ is a linear (super-)operator and ρ_t is the system state, a positive matrix. When designing or characterizing a quantum system, one is usually interested in the steady state ρ_∞ (such that ℒ ρ_∞ = 0), the first few excited states, and trajectories t↦ ρ_t. In finite dimension, ρ_t is an n× n matrix, ℒ thus typically costs n^4 to store explicitly as a dense matrix, and O(n^6) to diagonalize or invert exactly, making standard linear algebraic techniques expensive for large systems. However, ℒ usually costs only O(n^3) to apply. This makes iterative methods appealing, but they do not work without a good preconditioner. In this article, our main observation is that a part of the Lindblad equation, corresponding to the so-called no-jump evolution 𝒮, can be inverted efficiently. Using this inverse map, we introduce an auxiliary completely positive trace-preserving (CPTP) map Φ whose fixed point is directly related to ρ_∞, all the other eigenvalues having smaller magnitude. The map Φ is thus well suited to iterative methods, and ρ_∞ can be found in a few Arnoldi iterations. Using the same inverse map 𝒮^−1 as preconditioner, we compute the low-lying spectrum efficiently via shift-invert Arnoldi, and, as a proof of concept, build an implicit time integrator that is competitive on stiff systems in the low-precision regime. For the steady-state and low excited states problems, our methods scale like O(n^3) per iteration and offer state-of-the-art performance on CPU and GPU.
ER -