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Polynomial-time thermalization and Gibbs sampling from system-bath couplings

Samuel Slezak, Matteo Scandi, Álvaro M. Alhambra, Daniel Stilck França, Cambyse Rouzé

TL;DR

The paper addresses how quickly Lindbladian open-system dynamics converge to Gibbs states, focusing on two physically motivated models: repeated-interaction Gibbs sampling and macroscopic-bath thermalization. It introduces a gap-extrapolation lemma that transfers spectral-gap lower bounds from quasi-local to non-local Lindbladians, enabling polynomial-time convergence proofs beyond commuting or strictly local settings. The main results show polynomial-time Gibbs-state preparation for high-temperature local lattices, weakly interacting fermions, and 1D spin chains via RI sampling, as well as fast thermalization for MB dynamics at high temperature and commuting models like the 2D Toric code; precise time bounds are provided in terms of system size, temperature, and gap constants. This work provides theoretical support for near-term dissipative quantum state-preparation methods and establishes a broadly applicable technique for analyzing fast mixing in non-local quantum Markov processes, with potential implications for quantum simulations and quantum memories.

Abstract

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.

Polynomial-time thermalization and Gibbs sampling from system-bath couplings

TL;DR

The paper addresses how quickly Lindbladian open-system dynamics converge to Gibbs states, focusing on two physically motivated models: repeated-interaction Gibbs sampling and macroscopic-bath thermalization. It introduces a gap-extrapolation lemma that transfers spectral-gap lower bounds from quasi-local to non-local Lindbladians, enabling polynomial-time convergence proofs beyond commuting or strictly local settings. The main results show polynomial-time Gibbs-state preparation for high-temperature local lattices, weakly interacting fermions, and 1D spin chains via RI sampling, as well as fast thermalization for MB dynamics at high temperature and commuting models like the 2D Toric code; precise time bounds are provided in terms of system size, temperature, and gap constants. This work provides theoretical support for near-term dissipative quantum state-preparation methods and establishes a broadly applicable technique for analyzing fast mixing in non-local quantum Markov processes, with potential implications for quantum simulations and quantum memories.

Abstract

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.
Paper Structure (24 sections, 24 theorems, 167 equations, 1 figure)

This paper contains 24 sections, 24 theorems, 167 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Repeated interaction models.
  4. Macroscopic baths
  5. Main Results
  6. Polynomial time Gibbs sampling via repeated interactions.
  7. Efficient physical thermalization
  8. Faster thermalization for commuting models
  9. Methods
  10. Repeated interaction Gibbs sampling.
  11. Efficient physical thermalization.
  12. Discussion and Conclusion
  13. KMS Symmetric Lindbladians and mixing times
  14. Spectral gaps and fast mixing
  15. Modified logarithmic Sobolev inequality and rapid mixing
  16. ...and 9 more sections

Key Result

Theorem 1

Repeated interaction Gibbs sampling prepares a state $\epsilon$ close in $1$-norm to the Gibbs state $\rho_\beta^S$ with total Hamiltonian simulation time: for sufficiently small $\beta \le \beta^*= \mathcal{O}(1)$ for $(k,l)$-local Hamiltonians with maximum interaction strength $h$ as in rouzé2024efficientthermalizationuniversalquantum, and at any constant $\beta$ for local weakly interacting Fe

Figures (1)

  • Figure 1: Representations of the system-bath couplings we consider. In a), each step of the algorithm couples a single qubit bath with a random frequency $\omega$ to a randomly chosen qubit with the interaction $V=A^a\otimes X$ with time dependent strength $\alpha f_\kappa(t)$, resulting in the jump operators $\widehat{A}_\kappa^a(\omega)$ in the Lindbladian with their locality is determined by the width of the function $f_\kappa$. In b) a bath of qubits is coupled to the system via the interaction $V=\sum_a A^a\otimes B^a$ with strength $\alpha$, leading to jump operators $\widehat{A}^a_\alpha(\omega)$ with their locality being determined by the bath correlation functions and the coupling strength $\alpha$.

Theorems & Definitions (39)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • Lemma 2
  • Lemma 4
  • Lemma 5
  • proof
  • proof : Proof of Lemma \ref{['lem:CKG-monotonicity']}.
  • Lemma 3
  • ...and 29 more