Exponential mixing time for the Davies dynamics of the quantum random energy model

Abstract

We prove exponentially slow thermalization for the quantum random energy model (QREM) under Davies dynamics. The QREM is the $n$-qubit Hamiltonian $H=\Gamma T+U$, where $U$ assigns independent Gaussian energies with variance equal to the system size to computational-basis configurations and $T$ is the transverse field generated by single-qubit Pauli $X$ operators. Let ${\beta }_c=\sqrt{2\log 2}$ be the freezing inverse temperature of the classical random energy model. For every inverse temperature $\beta >{\beta }_c$ and every transverse-field strength $0\le \Gamma <{\Gamma }_c(\beta )$, where $e^{\beta {\beta }_c}=2\cosh (\beta {\Gamma }_c(\beta ))$, we show that the mixing time of the dissipative Davies dynamics associated with single-qubit Pauli system--bath couplings grows exponentially with the system size, almost surely; the bound is unchanged if the standard zero-Bohr-frequency dephasing terms are included. The result is uniform over all Davies rates satisfying detailed balance and a polynomial upper bound and does not assume primitivity. In the spin-glass phase, the Gibbs weight is carried by a sparse family of deep classical configurations that are separated by an extensive Hamming distance. A localization theorem for the QREM [Manai, Warzel, Comm. Math. Phys. 402, 1259 (2023)] transfers this geometry to the corresponding low-energy quantum eigenstates. Consequently, the Gibbs weight of the intermediate spectral sector is exponentially small, while single-qubit bath operators have exponentially small matrix elements between the two macroscopic sides of the bottleneck. A Davies-specific flux estimate then converts these two facts into an exponential lower bound on the mixing time. This gives a rigorous open-system dynamical manifestation of the frozen, localized low-energy structure of the QREM throughout its equilibrium spin-glass phase below the quantum transition line.