Dephasing-Enhanced Response and Phase Transitions in Quantum Boltzmann Samplers

Dephasing can drive a quantum Boltzmann sampler into an ordered phase while its programmed probability distribution remains disordered. We trace this transition to an enhanced conditional response: coherence loss changes how each spin responds to its neighbors, and interactions amplify the change. For all-to-all coupling, we establish the stationary transition at finite dephasing. Near criticality, the magnetization distribution depends on the ratio of dephasing strength to the square root of system size, retaining a finite imprint of coherence even as the local coherent correction vanishes. In the strong-dephasing limit on a square lattice, the dynamics generates effective multispin interactions and probability currents, with Ising-like critical statistics. Adjusting the local dissipative update restores the target probability process on arbitrary graphs.