Boosting thermalization of classical and quantum many-body systems

TL;DR

The paper addresses the challenge of efficiently preparing thermal states in classical and quantum many-body systems by engineering Lindbladians whose spectral gap governs relaxation. It introduces a transformation between finite- and infinite-temperature Lindbladians, leverages the dynamical Lie algebra and tensor-network methods for scalable construction, and shows that enforcing the symmetries of the thermal state reduces parameter count and enhances the gap via gradient-based optimization. It provides a variational framework and SDP-based lower bounds to certify relaxation rates, demonstrates substantial gap improvements in 1D Ising models with single- and multi-spin flips, and applies the approach to both classical and quantum spin systems, including the transverse-field Ising model. The work advances practical thermal-state preparation and offers a path toward certified performance bounds for open quantum-many-body dynamics with potential impact on quantum simulation and quantum thermodynamics.

Abstract

Understanding and optimizing the relaxation dynamics of many-body systems is essential both for foundational studies in quantum thermodynamics and for applications such as quantum simulation and quantum computing. Efficient preparation of thermal states of a many-body Hamiltonian is governed by the spectral properties of the associated Lindbladian, in particular its spectral gap, which determines the slowest relaxation rate. In this work, we develop a systematic framework for constructing Lindbladians that prepare thermal states. Our approach reveals a simple relation between the relaxation dynamics at finite and infinite temperatures. The framework is scalable to larger system sizes when implemented using tensor-network methods. We find that efficient thermalization requires that the relaxation dynamics respect the symmetries of the thermal state, which reduces the number of free parameters. By applying gradient-based optimization to the Lindbladians, we enhance the spectral gap and thereby boost thermalization. When applied to both classical and quantum spin models, our method demonstrates a substantial enhancement of the spectral gap. For larger system sizes, our approach provides a variational upper bound and enables a certified lower bound on the minimum relaxation rate.

Figures (2)

• Figure 1: The normalized gap $N\Delta$ as a function of inverse temperature $\beta$ for the 1D kinetic Ising model, shown on (a) linear and (b) logarithmic scales. Circles correspond to the single-spin flip case, and squares to the case combining the single- and the double-spin flips. Dashed and solid curves show exact-diagonalization results for the canonical and optimized kinetic coefficients at $N=10$, while open and filled markers show the corresponding DMRG results at $N=40$. In the inset of panel (b), dashed and solid gray curves show the SDP-certified gaps for the single-spin flip case with the canonical and optimized kinetic coefficients, respectively. The markers show the DMRG results for $N=40$.
• Figure 2: The normalized gap $N\Delta$ as a function of $\beta J$ for the 1D transverse-field Ising model, shown on (a) linear and (b) logarithmic scales. Circles correspond to $g=0.5$, and squares to $g=1$. The spectral gaps in both cases are obtained from the parent Hamiltonian in the thermofield-double space. Dashed and solid curves represent exact-diagonalization results for the canonical and optimized kinetic coefficients at $N=5$, while open and filled markers denote the corresponding DMRG results at $N=20$.