Fig. 1: The Gibbs state preparation algorithm for linear spin chains.
a From the input Hamiltonian H2N of a 2N − spin site chain, b we can define the global Gibbs state of the system via its density of states \({\rho }_{2N}={e}^{-\beta {H}_{2N}}/{\mathcal{Z}}\), where β is the inverse temperature 1/kBT and \({\mathcal{Z}}\) is the partition function of the system. c A local cluster expansion is devised to prepare the Gibbs state. We begin with a half-cut expansion of the Gibbs state, going from ρ2N to ρN ⊗ ρN, leaving us with an expansion of order \({\mathcal{O}}(\beta )\). We define the cluster cumulant terms Δn as described in Eq. (2). They are considered to be classically simulatable as they are not mixed states. They are used to include the cross-boundary interaction terms missing from the expansion after the half-cut. d By writing out the expansion with both the half-cut approximation and the cross-boundary cumulant terms, we can obtain an expansion of higher accuracy in β (i.e., \({\mathcal{O}}({\beta }^{n})\), where n is the chain length of the largest Δn term included) without needing to sample from larger clusters. Different considerations would be needed in the case of going beyond 1D, as we would need to instead define the cumulant terms as ΔT(n), accounting for the different topologies T(n) that could be embedded across the boundary of the half-cut of said lattice.
