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 2q with bounded nearest-neighbor interactions, the algorithm prepares the Gibbs state from any initial state to trace-norm error 0<ε≤1/2 in expected total Hamiltonian evolution time O(n/ε). This improves the previously proved O(n⁶/ε) bound for single-ancilla system-bath Gibbs samplers, up to logarithmic factors. The proposed algorithm couples 4q-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 L² 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.