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

Technical Summary: Polynomial-time thermalization and Gibbs sampling from system-bath couplings

Problem Statement
The paper addresses the convergence speed of Lindbladian dynamics, which model open quantum systems weakly coupled to a thermal bath, towards their steady Gibbs states. Understanding these convergence rates (mixing times) is critical for bounding the runtimes of quantum Gibbs sampling algorithms and the timescales of physical thermalization.

Two specific families of processes are investigated:

1. Repeated-Interaction Gibbs Sampling: An algorithm where a system undergoes repeated weak interactions with ancillary thermal qubits. While conceptually simple and suitable for near-term implementation, previous performance guarantees were limited to commuting Hamiltonians.
2. Macroscopic Bath Thermalization: A model derived from weak system-bath interactions where the bath is a large, uncontrolled system. The resulting Lindbladian generators satisfy the Kubo-Martin-Schwinger (KMS) detailed balance condition. However, for the approximation to be precise (small coupling $\alpha$), the jump operators become highly non-local, rendering existing mixing time proofs inapplicable.

The core technical challenge is that existing proofs for spectral gaps (which govern convergence speed) of KMS-symmetric Lindbladians rely on the jump operators being quasi-local. In both models above, the regimes where the physical approximation is accurate (large $\kappa$ for repeated interactions, small $\alpha$ for macroscopic baths) result in non-local generators, creating a gap between physical accuracy and provable convergence.

Methodology
The authors introduce a novel technical lemma (Lemma 1) that establishes the monotonicity of spectral gaps for specific families of Lindbladians.

• For the repeated-interaction family $\{LG_\kappa\}$, the spectral gap $\lambda_{Gap}$ is non-decreasing with the parameter $\kappa$ (the width of the interaction function).
• For the macroscopic bath family $\{L^{MB}_\alpha\}$, the spectral gap is non-increasing with the coupling strength $\alpha$.

This monotonicity allows the authors to "bootstrap" spectral gap lower bounds. They can prove a gap exists for a regime where the operators are quasi-local (where standard proof techniques apply) and extrapolate that bound to the non-local regime where the physical models are accurate. This is achieved by expressing the Dirichlet forms of the Lindbladians as convolutions.

Key Contributions and Results

1. Efficient Preparation via Repeated Interactions (Theorem 1):
   The authors prove that the repeated-interaction Gibbs sampling algorithm prepares Gibbs states in polynomial time for several non-commuting systems, extending beyond the previously known commuting cases. The total Hamiltonian simulation time scales as $\tilde{O}(N^{10}/\epsilon^4)$ for:

   • High-temperature local qubit models.
   • Weakly interacting fermionic models at any constant temperature.
   • 1D spin chains at any constant temperature.
     This demonstrates that simple, early fault-tolerant schemes can efficiently prepare complex Gibbs states.

2. Efficient Physical Thermalization (Theorem 2):
   The paper proves that open many-body systems weakly coupled to high-temperature baths reach their Gibbs steady state in polynomial time. Specifically, for local Hamiltonians at high temperatures ($\beta \leq \beta^*$), the system state becomes $\epsilon$-close to the Gibbs state in 1-norm within time $t = \tilde{O}(N^7 \epsilon^{-2})$ with a coupling strength $\alpha = \tilde{O}(N^{-3}\epsilon)$. This provides a first-principles justification that KMS Lindbladians accurately capture natural thermal relaxation down to the steady state.

3. Faster Thermalization for Commuting Models (Theorem 3):
   For commuting Hamiltonians (specifically 2D Toric codes and related CSS codes), the authors combine their extrapolation technique with recent results on Modified Logarithmic Sobolev Inequalities (MLSI). They derive tighter bounds showing that these systems thermalize in time $t = \tilde{O}(\gamma_{max}^4 N^4 \tau^2 / \epsilon^2)$. This offers a dynamical justification for the claim that low-dimensional CSS codes cannot serve as good quantum memories.

Significance
The paper claims to provide theoretical footing for the early-fault tolerant dissipative preparation of complex quantum states, a direction that has seen recent experimental progress. By overcoming the locality barrier in spectral gap proofs, the authors demonstrate that:

• Simple dissipative quantum algorithms are capable of preparing complex Gibbs states efficiently.
• Lindblad dynamics derived from weak system-bath couplings accurately capture thermal relaxation timescales for non-commuting many-body systems.
• The monotonicity of spectral gaps is a powerful tool that can likely be applied to other Lindbladians currently out of reach of standard proof techniques, suggesting a qualitative similarity in mixing times across various proposed Gibbs samplers.

The work does not propose new experimental setups but rather validates the theoretical efficiency of existing algorithmic and physical models, bridging the gap between physical approximations and rigorous convergence guarantees.