Polynomial-Time Thermalization Achieves Gibbs Sampling for Complex Quantum Systems

Researchers are tackling a fundamental problem in physics: how quickly do complex systems reach thermal equilibrium and how efficiently can we sample from their probability distributions. Samuel Slezak (Univ Lyon, ENS Lyon, UCBL, CNRS, Inria, LIP), Matteo Scandi (Instituto de Física Téorica UAM/CSIC), and Álvaro M Alhambra (Instituto de Física Téorica UAM/CSIC), alongside colleagues such as D Stilck França and C Rouzé, demonstrate that certain dissipative processes , modelling both thermalisation and a repeated-interaction Gibbs sampling algorithm , converge to a steady state in polynomial time for a range of physically relevant systems, including spin chains and interacting fermions. This is significant because it proves simple algorithms can reliably prepare complex quantum states and validates the use of Lindblad dynamics in accurately describing thermal relaxation , offering crucial insights for bounding computational runtimes and understanding timescales in open quantum systems.

This is particularly significant as existing, more complex algorithms often require intricate implementations unsuitable for near-term quantum computers. Crucially, the work establishes that these Lindblad dynamics accurately capture thermal relaxation, validating their use as a reliable approximation of natural thermalization processes. Experiments show that the convergence speed of these Lindbladians is critical for bounding algorithmic runtimes and accurately determining thermalization timescales.

This advancement enables the analysis of systems where the interactions are not limited to nearby components, significantly broadening the scope of applicable models. This work extends beyond algorithmic applications, as the concept of KMS detailed balance, a key property ensuring convergence, is also relevant to understanding open-system thermalization. By strengthening arguments on the thermalization times of quantum memories using modified logarithmic Sobolev inequalities for commuting Hamiltonians, the study provides a comprehensive framework for analysing the efficiency of these quantum processes. The team believes this technique can be applied to demonstrate fast mixing for other Lindbladians currently beyond the reach of existing proof techniques, suggesting a qualitatively similar behaviour in their mixing times.

Extrapolating Lindbladian Spectral Gaps for Convergence Analysis

Scientists investigated the convergence rates of Lindbladian dynamics, crucial for determining the runtime of algorithms and thermalisation timescales. This breakthrough allowed them to analyse complex systems previously inaccessible to existing proof techniques, demonstrating the power of this new approach. Researchers rigorously analysed the mixing time of these Lindbladians, establishing performance guarantees for the repeated-interaction algorithm even with non-commuting Hamiltonians. This involved demonstrating that the algorithm efficiently prepares complex Gibbs states, extending previous results limited to commuting systems.

The team harnessed the Kubo-Martin-Schwinger (KMS) detailed balance property, ensuring convergence to the Gibbs state, and meticulously characterised the mixing time under various conditions, high temperatures, weak interactions, and 1D systems. This was achieved by analysing a model of thermalisation derived from weak system-bath interactions, confirming the accuracy of Lindbladian dynamics in capturing natural many-body thermalisation. The research confirms that these Lindbladians accurately capture the natural many-body thermalisation process, down to the steady state.

Polynomial Time Convergence of Dissipative Quantum Algorithms

Experiments revealed that these Lindbladian processes, approximating weakly coupled systems to a bath, accurately capture thermal relaxation and provide a means of bounding algorithmic runtimes and thermalization timescales. The team measured convergence speeds of these Lindbladians to their steady states, establishing crucial benchmarks for algorithmic performance. Results demonstrate that the Gibbs sampling algorithm achieves efficient preparation, extending performance guarantees to a wide range of physically relevant many-body settings. Measurements confirm that KMS Lindbladians accurately capture natural many-body thermalization, reaching Gibbs steady states quickly and accurately, as expected from physical principles. Scientists recorded that this extrapolation is crucial, as previous results on physically-generated Lindbladians were limited to commuting models and vanishing coupling with the bath. Tests prove that the underlying technique can be applied to a variety of Gibbs samplers, suggesting qualitatively similar mixing times across different implementations.

Toric Code Dynamics and Rapid Thermalisation reveal surprising

Specifically, for the two-dimensional toric code, the associated Davies dynamics exhibit a faster exponential decay rate, satisfying the modified logarithmic Sobolev inequality. This result provides the first complete dynamical justification for the long-standing claim that CSS codes in low lattice dimensions are unsuitable as quantum memories. The authors established a time complexity of t = e^O(γ^4 max(N^4τ^2) ε^2) for thermalization in the toric code with a coupling constant α = e^O(ε γ^2 min(N^2τ)). The authors acknowledge a limitation in that achieving ε-close approximation to the Gibbs state often requires parameters scaling polynomially with system size and inverse error tolerance, leading to extensive support for the jump operators. Future research directions include exploring the applicability of these findings to a wider range of physical systems and investigating the potential for optimising the parameters to reduce computational cost. These findings are significant as they establish a theoretical foundation for understanding the efficiency of dissipative algorithms in quantum simulation and provide insights into the dynamics of open quantum systems.