Digitized Counterdiabatic Quantum Sampling

We propose digitized counterdiabatic quantum sampling (DCQS), a hybrid quantum-classical algorithm for efficient sampling from energy-based models, such as low-temperature Boltzmann distributions. The method utilizes counterdiabatic protocols, which suppress non-adiabatic transitions, with an iterative bias-field procedure that progressively steers the sampling toward low-energy regions. We observe that the samples obtained at each iteration correspond to approximate Boltzmann distributions at effective temperatures. By aggregating these samples and applying classical reweighting, the method reconstructs the Boltzmann distribution at a desired temperature. We define a scalable performance metric, based on the Kullback-Leibler divergence and the total variation distance, to quantify convergence toward the exact Boltzmann distribution. DCQS is validated on one-dimensional Ising models with random couplings up to 124 qubits, where exact results are available through transfer-matrix methods. We then apply it to a higher-order spin-glass Hamiltonian with 156 qubits executed on IBM quantum processors. We show that classical sampling algorithms, including Metropolis-Hastings and the state-of-the-art low-temperature technique parallel tempering, require up to three orders of magnitude more samples to match the quality of DCQS, corresponding to an approximately 2x runtime advantage. Boltzmann sampling underlies applications ranging from statistical physics to machine learning, yet classical algorithms exhibit exponentially slow convergence at low temperatures. Our results thus demonstrate a robust route toward scalable and efficient Boltzmann sampling on current quantum processors.