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 $\widetilde{O}(\sqrt{N}/ε)$ at fixed inverse temperature and bounded single-particle bandwidth, while ground state preparation of gapped systems requires only $\mathrm{polylog}(N/ε)$ 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.