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arXiv:2512.03457

Beyond Lindblad Dynamics: Rigorous Guarantees for Thermal and Ground State Preservation under System Bath Interactions

Ke Wang, Zhiyan Ding

quant-phSubmitted Dec 3, 2025• Announced Dec 4, 2025• 0 citations• Revised Feb 16, 2026 (v2)
v1 · Dec 3, 2025v2 · Feb 16, 2026v3 · Jul 3, 2026v4 · Sep 19, 2026
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Abstract

We establish new theoretical results demonstrating the efficiency and robustness of system bath interaction models for quantum thermal and ground state preparation. Unlike prior analyses, which typically relies on the Lindblad limit and require vanishing coupling strengths $o(1)$, we rigorously show that efficient state preparation remains possible far beyond this regime, even when the coupling strength is $Θ(1)$. We first prove that even with constant coupling strength, the induced quantum channel still approximately fixes the target state. For thermal state preparation, we then develop a general perturbative framework that yields end to end complexity bounds outside weak coupling, and in particular proves that the mixing time scales as the inverse square of the coupling strength. This framework extends to broad Hamiltonian for which KMS detailed balance Lindbladians are known to mix. These bounds substantially improve upon prior results, and numerical simulations further confirm the robustness of the system bath interaction framework across both weak and strong coupling regimes.

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We establish new theoretical results demonstrating the efficiency and robustness of system bath interaction models for quantum thermal and ground state preparation. Unlike prior analyses, which typically relies on the Lindblad limit and require vanishing coupling strengths o(1)o(1)o(1), we rigorously show that efficient state preparation remains possible far beyond this regime, even when the coupling strength is Θ(1)Θ(1)Θ(1). We first prove that even with constant coupling strength, the induced quantum channel still approximately fixes the target state. For thermal state preparation, we then develop a general perturbative framework that yields end to end complexity bounds outside weak coupling, and in particular proves that the mixing time scales as the inverse square of the coupling strength. This framework extends to broad Hamiltonian for which KMS detailed balance Lindbladians are known to mix. These bounds substantially improve upon prior results, and numerical simulations further confirm the robustness of the system bath interaction framework across both weak and strong coupling regimes.
We investigate quantum thermal state preparation algorithms based on system-bath interactions and uncover a surprising phenomenon in the weak-coupling regime. We rigorously prove that, if the system-bath interaction is engineered so that the transition part of the approximate Lindbladian generator satisfies the Kubo--Martin--Schwinger (KMS) detailed balance condition, then the unique fixed point of the dynamics can be made arbitrarily close to the Gibbs state in the weak-coupling limit, regardless of the structure of the Lamb shift term. Importantly, this remains true even when the approximate Lindbladian differs substantially from the ideal Davies generator and the Lamb shift term does not commute with the thermal state. Our result shows that the role of the KMS detailed balance condition extends well beyond standard Lindbladian dynamics, serving as a general principle for a broader class of dissipative systems. Furthermore, by combining this with a general perturbation framework, we bound the mixing time of the dynamics and establish an end-to-end complexity of O(ε−1)\mathcal{O}(\varepsilon^{-1})O(ε−1) for Gibbs state preparation. These guarantees apply to any Hamiltonian whose associated KMS-detailed-balance Lindbladian is known to be fast mixing.
Thermal state preparation is a central challenge in the simulation of quantum many-body systems. Yet, provably efficient algorithms for this task were only introduced recently [Chen et al. Nature 646, 561 (2025)]. These algorithms are based on dissipative Lindbladian evolution which exactly fixes the thermal state. Controlled and efficient digital simulation of this evolution, although possible in principle, remains out of reach for present-day quantum hardware. Subsequent work has therefore focused on analog approximations of the proposed Lindbladians via `collision models' with relatively modest requirements -- a resettable bath of ancilla qubits whose couplings to the system can be tuned in time-dependent fashion -- while still admitting rigorous fixed-point error bounds. Existing rigorous approaches, however, do not exploit the fact that these constructions generically implement not only the desired Lindblad dynamics, but also an additional unitary evolution generated by the system Hamiltonian which may aid convergence to the thermal state [Lloyd and Abanin arXiv:2506.21318 (2025)]. Here, we show that this unitary contribution does indeed tighten the fixed-point error bound and demonstrate that it is rigorously controlled by the system-bath coupling strength JJJ, scaling as J2J^2J2. This demonstrates that the effect of the spurious `Lamb shift' term generated by the system-bath interaction can be controlled by tuning JJJ. We clarify the role, previously observed, of a randomized implementation in suppressing possible resonances of the drive with the many-body spectrum, and bound the additional variance that this randomization imposes on observables. Finally, we numerically study aspects of the protocol which are relevant for its practical realization, such as the mixing time.
Learning quantum interactions from finite-temperature many-body systems is a central task in emerging quantum platforms. Recently, the problem of learning from lattice quantum Gibbs states has found rigorous, efficient protocols. Nevertheless, exact Gibbs states, as the input premise, are in fact computationally intractable to prepare and may not faithfully represent generic finite-temperature quantum systems. In contrast, a system coupled to a heat bath can be stuck at an approximate stationary state (metastable state) long before it truly equilibrates. Here, we formulate a physically and algorithmically consistent alternative: learning from such metastable states of detailed-balanced master equations (Lindbladians) arising from system-bath interactions. We distill the algorithmic mechanism and structural condition underlying Gibbs-state learning and extend it in full to metastable states, attaining nearly optimal sample and computational complexity (in the system size and the precision). More broadly, we sharpen notions of metastability and develop a unified framework for finite-temperature learning.
Many physical phenomena, including thermalization in open quantum systems and quantum Gibbs sampling, are modeled by Lindbladians approximating a system weakly coupled to a bath. Understanding the convergence speed of these Lindbladians to their steady states is crucial for bounding algorithmic runtimes and thermalization timescales. We study two such families of processes: one characterizing a repeated-interaction Gibbs sampling algorithm, and another modeling open many-body quantum thermalization. We prove that both converge in polynomial time for several non-commuting systems, including high-temperature local lattices, weakly interacting fermions, and 1D spin chains. These results demonstrate that simple dissipative quantum algorithms can prepare complex Gibbs states and that Lindblad dynamics accurately capture thermal relaxation. Our proofs rely on a novel technical result that extrapolates spectral gap lower bounds from quasi-local Lindbladians to the non-local generators governing these dynamics.
Abstract We propose new quantum algorithms for thermal and ground state preparation based on system-bath interactions. These algorithms require only forward evolution under a system-bath Hamiltonian in which the bath is a single reusable ancilla qubit, making them especially well-suited for early fault-tolerant quantum devices. By carefully designing the bath and interaction Hamiltonians, we prove that the fixed point of the dynamics accurately approximates the desired quantum state. Furthermore, we establish theoretical guarantees on the mixing time, and thereby providing a rigorous justification for the end-to-end efficiency of system-bath interaction models in thermal and ground state preparation, for several physically relevant systems.
Away from equilibrium, the properties of open quantum systems depend on the details of their environment. A microscopic derivation of a master equation (ME) is therefore crucial. Of particular interest are Lindblad-type equations, not only because they provide the most general class of Markovian MEs, but also since they are the starting point for efficient quantum trajectory simulations. Lindblad-type MEs are commonly derived from the Born-Markov-Redfield equation via a rotating-wave approximation (RWA). However the RWA is valid only for ultraweak system-bath coupling and often fails to accurately describe nonequilibrium processes. Here we derive an alternative Lindbladian approximation to the Redfield equation, which does not rely on ultraweak system-bath coupling. Applying it to an extended Hubbard model coupled to Ohmic baths, we show that, especially away from equilibrium, it provides a good approximation in large parameter regimes where the RWA fails.