Lindblad engineering for quantum Gibbs state preparation under the eigenstate thermalization hypothesis

Eric Brunner; Luuk Coopmans; Gabriel Matos; Matthias Rosenkranz; Frederic Sauvage; Yuta Kikuchi

doi:10.22331/q-2025-08-29-1843

Lindblad engineering for quantum Gibbs state preparation under the eigenstate thermalization hypothesis

Eric Brunner¹, Luuk Coopmans¹, Gabriel Matos^1,2, Matthias Rosenkranz¹, Frederic Sauvage¹, and Yuta Kikuchi^3,4

¹Quantinuum, Partnership House, Carlisle Place, London SW1P 1BX, United Kingdom
²Quantinuum, 17 Beaumont St., Oxford OX1 2NA, United Kingdom
³Quantinuum K.K., Otemachi Financial City Grand Cube 3F, 1-9-2 Otemachi, Chiyoda-ku, Tokyo, Japan
⁴Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS), RIKEN, Wako, Saitama 351-0198, Japan

Published:	2025-08-29, volume 9, page 1843
Editor:	Álvaro Alhambra
Eprint:	arXiv:2412.17706v3
Doi:	https://doi.org/10.22331/q-2025-08-29-1843
Citation:	Quantum 9, 1843 (2025).

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Abstract

Building upon recent progress in Lindblad engineering for quantum Gibbs state preparation algorithms, we propose a simplified protocol that is shown to be efficient under the eigenstate thermalization hypothesis (ETH). The ETH reduces circuit overheads of the Lindblad simulation algorithm and ensures a fast convergence toward the target Gibbs state. Moreover, we show that the realized Lindblad dynamics exhibits an inherent resilience against stochastic noise, opening up the path to a first demonstration on quantum computers. We complement our claims with numerical studies of the algorithm's convergence in various regimes of the mixed-field Ising model. In line with our predictions, we observe a mixing time scaling polynomially with system size when the ETH is satisfied. In addition, we assess the impact of algorithmic and hardware-induced errors on the algorithm's performance by carrying out quantum circuit simulations of our Lindblad simulation protocol with a local depolarizing noise model. This work bridges the gap between recent theoretical advances in dissipative Gibbs state preparation algorithms and their eventual quantum hardware implementation.

Featured image: Trace distance between the target Gibbs state $\sigma_\beta$ and the state $\rho(t)$ evolved under Lindblad dynamics of the mixed-field Ising model with single random jump operators at each time step for $n=3,\dots, 8$ qubits and $\beta=(2J)^{-1}$. We consider a set of size $|A| = 20$ of random $2$-local Pauli jump operators and the Hamiltonian parameter points REG (left, dominated by the longitudinal field) and CH (right, quantum chaotic regime). Thin lines are individual trajectories of the randomized Lindblad simulation protocol, thick lines are trace distances between the Gibbs state and the time-evolved state averaged over trajectories. Vertical lines indicate our mixing time estimate (where the trace distance drops below $10^{-2}$. The results confirm much faster mixing in the quantum chaotic regime (CH) than in the non-chaotic regime (REG). On the other hand, the system in parameter regime REG converges to a state closer to the Gibbs state.

Popular summary

Quantum Gibbs states play an important role as a resource for many quantum algorithms and as a tool in the search for quantum computational advantage. Recently, there has been remarkable progress in quantum Gibbs state preparation algorithms based on the simulation of dissipative quantum dynamics. Despite many theoretical works, the feasibility of these algorithms on noisy quantum computers has barely been explored, and comprehensive numerical studies of their convergence are rare.

In our work we address this topic and make important contributions towards the first demonstration of this class of algorithms on quantum hardware. First, we propose a simplified circuit construction leading to significantly reduced quantum resource requirements compared to previous proposals, and proof a fast (polynomial-time) convergence towards the target Gibbs state under the eigenstate thermalization hypothesis. Second, we show that our protocol (and comparable dissipative Gibbs state preparation algorithms) exhibits an inherent resilience against stochastic noise. These are critical observations for realizing future hardware experiments.

Our study combines established techniques in quantum chaos with the mixing time analysis of dissipative quantum systems. We perform extensive numerical studies of the mixing time and convergence accuracy for a 1D mixed-field Ising model in chaotic and non-chaotic parameter regimes, and confirm that chaoticity of the Hamiltonian leads to faster convergence to the Gibbs state. Moreover, we quantify the trade-off between algorithmic errors and the impact of hardware noise via circuit simulations under a local depolarizing noise model. Such practical studies are crucial to assess the feasibility of dissipative Gibbs state preparation methods on noisy hardware.

► BibTeX data

@article{Brunner2025lindbladengineering,
  doi = {10.22331/q-2025-08-29-1843},
  url = {https://doi.org/10.22331/q-2025-08-29-1843},
  title = {Lindblad engineering for quantum {G}ibbs state preparation under the eigenstate thermalization hypothesis},
  author = {Brunner, Eric and Coopmans, Luuk and Matos, Gabriel and Rosenkranz, Matthias and Sauvage, Frederic and Kikuchi, Yuta},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1843},
  month = aug,
  year = {2025}
}

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