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Well-conditioned iterative methods for large open quantum systems — arXiv:2608.30860

Source: arXiv (en)
Original: https://arxiv.org/abs/2608.30860
License (as archived): CC0-1.0
Archived: 2026-10-17T09:28:43.059Z
SHA-256 of the archived text: 4d03573fd210d0eb404d47ec337aef393723740deb8c26d3bd712649358474ce
Stable URL of this page: https://metraipolis.com/documento/dc_b024ee02a759

How to cite

Well-conditioned iterative methods for large open quantum systems — arXiv:2608.30860 — arXiv (en), https://arxiv.org/abs/2608.30860 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_b024ee02a759 (Metraipolis archive dc_b024ee02a759, sha256 4d03573fd210d0eb404d47ec337aef393723740deb8c26d3bd712649358474ce)

BibTeX

@misc{dcb024ee02a759,
  title = {Well-conditioned iterative methods for large open quantum systems — arXiv:2608.30860},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2608.30860},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_b024ee02a759; sha256 4d03573fd210d0eb404d47ec337aef393723740deb8c26d3bd712649358474ce},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2608.30860. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Markovian open quantum systems are well modeled by the Lindblad Master Equation (ME) $\frac{\mathrm{d}}{\mathrm{d} t} \rho_t = \mathcal{L} \rho_t$, where $\mathcal{L}$ is a linear (super-)operator and $\rho_t$ is the system state, a positive matrix. When designing or characterizing a quantum system, one is usually interested in the steady state $\rho_\infty$ (such that $\mathcal{L} \rho_\infty = 0$), the first few excited states, and trajectories $t\mapsto \rho_t$. In finite dimension, $\rho_t$ is an $n\times n$ matrix, $\mathcal{L}$ 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, $\mathcal{L}$ 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 $\mathcal{S}$, can be inverted efficiently. Using this inverse map, we introduce an auxiliary completely positive trace-preserving (CPTP) map $\Phi$ whose fixed point is directly related to $\rho_\infty$, all the other eigenvalues having smaller magnitude. The map $\Phi$ is thus well suited to iterative methods, and $\rho_\infty$ can be found in a few Arnoldi iterations. Using the same inverse map $\mathcal{S}^{-1}$ 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.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-01 in math.NA).
Categorie arXiv: Quantum Physics (quant-ph) ; Numerical Analysis (math.NA).
Pagina del lavoro: https://arxiv.org/abs/2608.30860 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-17T08:59:54.382Z dalla pagina https://arxiv.org/catchup/math.NA/2026-09-01?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).