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arXiv:2611.03422

Efficient thermalization of one-dimensional noncommuting Hamiltonians via system-bath interactions with $Θ(n)$ ancilla qubits

Zhiyan Ding, Bowen Li, Ruizhe Zhang

quant-phSubmitted Nov 2, 2026• Announced Nov 3, 2026
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Abstract

We prove rapid convergence of a system-bath interaction algorithm for preparing Gibbs states of one-dimensional noncommuting Hamiltonians at every fixed positive temperature. For a chain of $n$ qudits of local dimension $2^q$ with bounded nearest-neighbor interactions, the algorithm prepares the Gibbs state from any initial state to trace-norm error $0<\varepsilon\le1/2$ in expected total Hamiltonian evolution time $\widetilde{O}(n/\varepsilon)$. This improves the previously proved $\widetilde{O}(n^6/\varepsilon)$ bound for single-ancilla system-bath Gibbs samplers, up to logarithmic factors. The proposed algorithm couples $4^q-1$ resettable bath qubits to each site in parallel and randomizes the interaction time. Our analysis bounds both the one-round approximation error and the stationarity defect of an approximate stationary state in Gibbs-weighted $L^2$ norms, with only linear dependence on the system size. It also uses the dephasing induced by randomized evolution to control the extensive Lamb shift without treating it as a small perturbation of the dissipative dynamics.

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Efficient thermalization of one-dimensional noncommuting Hamiltonians via system-bath interactions with Θ(n)Θ(n)Θ(n) ancilla qubits
We prove rapid convergence of a system-bath interaction algorithm for preparing Gibbs states of one-dimensional noncommuting Hamiltonians at every fixed positive temperature. For a chain of nnn qudits of local dimension 2q2^q2q with bounded nearest-neighbor interactions, the algorithm prepares the Gibbs state from any initial state to trace-norm error 0<ε≤1/20<\varepsilon\le1/20<ε≤1/2 in expected total Hamiltonian evolution time O~(n/ε)\widetilde{O}(n/\varepsilon)O(n/ε). This improves the previously proved O~(n6/ε)\widetilde{O}(n^6/\varepsilon)O(n6/ε) bound for single-ancilla system-bath Gibbs samplers, up to logarithmic factors. The proposed algorithm couples 4q−14^q-14q−1 resettable bath qubits to each site in parallel and randomizes the interaction time. Our analysis bounds both the one-round approximation error and the stationarity defect of an approximate stationary state in Gibbs-weighted L2L^2L2 norms, with only linear dependence on the system size. It also uses the dephasing induced by randomized evolution to control the extensive Lamb shift without treating it as a small perturbation of the dissipative dynamics.
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.
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.
In this work we investigate a model of thermalization wherein a single ancillary qubit randomly interacts with the system to be thermalized. This not only sheds light on the emergence of Gibbs states in nature, but also provides a routine for preparing arbitrary thermal states on a digital quantum computer. For desired βββ and random interaction GGG the routine boils down to time independent Hamiltonian simulation and is represented by the channel Φ:ρ↦EGTrEnv[e−i(H+αG)t(ρ⊗e−βHEZ)ei(H+αG)t]Φ: ρ\mapsto \mathbb{E}_G {\rm Tr}_{\rm Env} \left[ e^{-i(H + αG)t} \left(ρ\otimes \frac{e^{-βH_E}}{\mathcal{Z}}\right) e^{i (H + αG)t} \right]Φ:ρ↦EG​TrEnv​[e−i(H+αG)t(ρ⊗Ze−βHE​​)ei(H+αG)t]. We rigorously prove that these dynamics reduce to a Markov chain process in the weak-coupling regime with the thermal state as the approximate fixed point. We upper bound the total simulation time required in terms of the Markov chain spectral gap λ⋆λ_\starλ⋆​, which we compute exactly in the ground state limit. These results are independent of any eigenvalue knowledge of the system, but we are further able to show that with knowledge of eigenvalue differences λS(i)−λS(j)λ_S(i) - λ_S(j)λS​(i)−λS​(j), then the total simulation time is dramatically reduced. The ratio of the complete ignorance simulation cost to the perfect knowledge simulation cost scales as O~(∥HS∥7δmin7ε3.5λ⋆(β)3.5)\widetilde{O} \left({\frac{\|{H_S}\|^7}{δ_{\rm min}^7 ε^{3.5} λ_\star(β)^{3.5}}}\right)O(δmin7​ε3.5λ⋆​(β)3.5∥HS​∥7​), where δmin⁡δ_{\min}δmin​ is related to the eigenvalue differences of the system. Additionally, we provide more specific results for single qubit and harmonic oscillator systems as well as numeric experiments with hydrogen chains. In addition to the algorithmic merits, these results can be viewed as broad extensions of the Repeated Interactions model to generic Hamiltonians with unknown interactions, giving a complete picture of the thermalization process for quantum systems.
Dissipative quantum algorithms for state preparation in many-body systems are increasingly recognised as promising candidates for achieving large quantum advantages in application-relevant tasks. Recent advances in algorithmic, detailed-balance Lindbladians enable the efficient simulation of open-system dynamics converging towards desired target states. However, the overall complexity of such schemes is governed by system-size dependent mixing times. In this work, we analyse algorithmic Lindbladians for Gibbs state preparation and prove that they exhibit rapid mixing, i.e., convergence in time poly-logarithmic in the system size. We first establish this for non-interacting spin systems, free fermions, and free bosons, and then show that these rapid mixing results are stable under perturbations, covering weakly interacting qudits and perturbed non-hopping fermions. Further, we adapt the techniques from separable qudits to the fermionic setting and prove rapid mixing of the strongly-interacting regime of the Fermi-Hubbard model, for which we also explicitly evaluate the guaranteed parameter regimes. Our results constitute the first efficient mixing bounds for non-commuting qudit models and bosonic systems at arbitrary temperatures. Compared to prior spectral-gap-based results for fermions, we achieve exponentially faster mixing, further featuring explicit constants on the maximal allowed interaction strength. This not only improves the overall polynomial runtime for quantum Gibbs state preparation, but also enhances robustness against noise. Our analysis relies on oscillator norm techniques from mathematical physics, where we introduce tailored variants adapted to specific Lindbladians $\unicode{x2014}$ an innovation that we expect to significantly broaden the scope of these methods.
It is of great interest to understand the thermalization of open quantum many-body systems, and how quantum computers are able to efficiently simulate that process. A recently introduced dissipative evolution, inspired by existing models of open system thermalization, has been shown to be efficiently implementable on a quantum computer. Here, we prove that, at high enough temperatures, this evolution reaches the Gibbs state in time scaling logarithmically with system size. The result holds for Hamiltonians that satisfy the Lieb-Robinson bound, such as local Hamiltonians on a lattice, and includes long-range systems. To the best of our knowledge, these are the first results rigorously establishing the rapid mixing property of high-temperature quantum Gibbs samplers, which is known to give the fastest possible speed for thermalization in the many-body setting. We then apply our result to the problem of estimating partition functions at high temperature, showing an improved performance over previous classical and quantum algorithms.
Classical Dobrushin--Shlosman theory uses heat-bath block updates to identify sharp temperature thresholds for establishing the rapid mixing of Gibbs samplers. In particular, these mixing guarantees often hold even when stricter single-site conditions fail. We extend this approach to noncommuting quantum lattice systems through a quantum Dobrushin--Shlosman condition for finite-block dynamics, which we call smoothed heat-bath dynamics. We establish this condition in two regimes. For finite-range quantum spin chains at every fixed finite temperature, we prove logarithmic trace-norm mixing in system size, with normalized bounds uniform in on-site fields and single exponential in inverse temperature. We also prove stability under small local perturbations of interacting, possibly noncommuting reference Hamiltonians on graphs of polynomial volume growth. For classical references with bounded local interactions, uniform strong spatial mixing implies our quantum condition. With bounded local terms and fixed physical, geometric and block parameters, the resulting samplers prepare Gibbs states to trace-norm error ε\varepsilonε using Npolylog⁡(N/ε)N\operatorname{polylog}(N/\varepsilon)Npolylog(N/ε) gates and classical operations. Our block updates combine a local Gibbs reset with quantum belief propagation to incorporate interactions across the block boundary, yielding exactly Gibbs-preserving, KMS-reversible channels. Two independently tunable parameters control contraction: an internal evolution time suppresses the influence of sites inside a block, while the block size allows interior contraction to dominate boundary influence. This finite-time relaxation has no counterpart in a classical heat-bath block update.