Sublinear thermal state and ground state preparation for free fermions via system-bath interaction

Dissipative algorithms for preparing thermal and ground states typically require a total Hamiltonian simulation time that grows at least linearly with system size N, even when their mixing time grows only polylogarithmically. We achieve sublinear simulation time for free fermions by engineering local interactions with a bath whose size grows in proportion to the system size. Starting from arbitrary initial states, our thermal state preparation algorithm achieves trace-norm error $ε$ in total simulation time O(√N/ε) at fixed inverse temperature and bounded single-particle bandwidth, while ground state preparation of gapped systems requires only polylog(N/ε) time. We further establish lower bounds for thermal state preparation within the bath protocols considered, identifying two obstacles to faster preparation, namely fourth-order deviations from the target thermal state and fluctuations induced by a random frequency shared by all bath qubits. Our results demonstrate that the linear simulation time barrier in dissipative state preparation can be overcome and clarify the bath and filter-design constraints that govern further complexity improvements.