Beyond Lindblad Dynamics: Rigorous Guarantees for Thermal and Ground State Preservation under System Bath Interactions

TL;DR

This paper proves that quantum state preparation via system–bath interactions can achieve accurate thermal and ground states beyond the conventional weak-coupling Lindblad regime, provided the cumulative coupling strength remains $\Theta(1)$. The core approach controls all orders of the Dyson expansion and analyzes multidimensional operator Fourier transforms to bound the induced channel's fixed point relative to the target state, with $\alpha=\Theta(\sigma^{-1/2})$ and $T=\widetilde{\Omega}(\sigma)$ yielding arbitrarily good approximations when the rescaled mixing time $t_{\text{mix}}=\alpha^2\tau_{\text{mix}}$ stays bounded. For thermal states, the fixed point error scales as $\mathcal{O}\big( (\beta\alpha^2/\sigma)\tau_{\text{mix}} \big)$, while ground-state guarantees require a spectral gap $\Delta$ and yield $\|\rho_{\text{fix}}(\Phi_{\alpha})-\sigma_β\|_1\le\varepsilon$ under appropriate $\sigma$ and $T$ choices, removing dependence on $\mathrm{poly}(\epsilon)$ in iteration counts under certain conditions. Numerically, TFIM and Hubbard models show robust convergence across both weak and strong coupling, with the spectral gap scaling as $\alpha^2$ and strong-coupling regimes performing even better than theory predicts, suggesting practical resilience and potential advantages for near-term quantum devices.

Abstract

We establish new theoretical results demonstrating the efficiency and robustness of system bath interaction models for quantum thermal and ground state preparation. Unlike existing analyses, which relies on the weak coupling Lindblad limit and require $O(ε)$ coupling strengths for $ε$ accuracy, leading to slow mixing, we rigorously show that accurate state preparation remains possible far beyond this regime. In particular, even when the cumulative coupling strength remains constant rather than vanishing, the induced quantum channel still approximately fixes the target state. Our proof introduces new techniques for controlling all orders of the Dyson expansion and for analyzing the associated multidimensional operator Fourier transforms. These bounds substantially improve upon prior results, and numerical simulations on the TFIM and Hubbard models further confirm the robustness of the system bath interaction framework across both weak and strong coupling regimes.

Figures (11)

• Figure 1: Numerical results for thermal state preparation of TFIM with $L=4$ sites in the regime $\alpha\sqrt{\sigma}=\Theta(1)$. We set $\beta=1$, $\sigma=2,4,8$, $\sqrt{2}\alpha=1, 0.5, 0.25, 0.1, 0.05, 0.01$, and $T=5\sigma$. The frequency $\omega$ is uniformly sampled from the interval $[0,5]$. Upper: evolution of fidelity; Lower: spectral gap of $\Phi_{\alpha}$.
• Figure 2: Numerical results for ground state preparation of the Hubbard model with $L=2$ sites in the regime $\alpha\sqrt{\sigma}=\Theta(1)$. We set $\alpha=0.5, 0.25, 0.125, 0.05, 0.025, 0.005$, $\sigma=4,8,16$, $T=5\sigma$, and also sample $\omega$ uniformly from $[0,5]$. Upper: evolution of energy, $\lambda_0$ is the ground state energy; Lower: spectral gap of $\Phi_{\alpha}$.
• Figure 3: Evolution of infidelity for thermal state preparation with $\alpha/\sqrt{\sigma}=\Theta(1)$. Upper: TFIM with $L=8$ sites; Lower: Hubbard model with $L=4$ sites.
• Figure 4: TFIM-4 with $\alpha\sqrt{\sigma}=\Omega(1)$ and $\beta=1$.
• Figure 5: TFIM-4 with $\alpha\sqrt{\sigma}=\Omega(1)$ and $\beta=\infty$.

Theorems & Definitions (18)

• Theorem 2: Informal: thermal state
• Theorem 3: Informal: ground state
• Definition 4
• Theorem 5
• proof
• Theorem 6: Thermal state, rigorous version
• Lemma 7
• proof : Proof of \ref{['lm:Tinf']}
• Lemma 8
• Lemma 8