Efficient thermalization of one-dimensional noncommuting Hamiltonians via system-bath interactions with $Θ(n)$ ancilla qubits

We prove rapid convergence of a system-bath interaction algorithm for preparing Gibbs states of one-dimensional noncommuting Hamiltonians at every fixed positive temperature. For a chain of $n$ qudits of local dimension $2^q$ with bounded nearest-neighbor interactions, the algorithm prepares the Gibbs state from any initial state to trace-norm error $0<\varepsilon\le1/2$ in expected total Hamiltonian evolution time $\widetilde{O}(n/\varepsilon)$. This improves the previously proved $\widetilde{O}(n^6/\varepsilon)$ bound for single-ancilla system-bath Gibbs samplers, up to logarithmic factors. The proposed algorithm couples $4^q-1$ resettable bath qubits to each site in parallel and randomizes the interaction time. Our analysis bounds both the one-round approximation error and the stationarity defect of an approximate stationary state in Gibbs-weighted $L^2$ norms, with only linear dependence on the system size. It also uses the dephasing induced by randomized evolution to control the extensive Lamb shift without treating it as a small perturbation of the dissipative dynamics.