Efficient quantum Gibbs sampling of stabilizer codes using hybrid computation

We present hybrid Gibbs sampling algorithms for the stabilizer code Hamiltonians of the rotated surface code and the toric code with only local quantum algorithms, using $\sim L\mathrm{/}2$ quantum circuit depth to prepare the Gibbs state of the rotated surface code Hamiltonian, and $L$ quantum circuit depth to prepare the Gibbs state of the toric code Hamiltonian, being $L$ the side of the side of the square lattice. We further show that if we allow for non-local gates, the Gibbs state of the periodic 1D Ising model can be prepared in logarithmic depth and linearly many simultaneous measurements.