Quantum Simulation of Classical Thermal States

We establish a connection between ground states of local quantum Hamiltonians and thermal states of classical spin systems. For any discrete classical statistical mechanical model in any spatial dimension, we find an associated quantum state such that the reduced density operator behaves as the thermal state of the classical system. We show that all these quantum states are unique ground states of a universal 5-body local quantum Hamiltonian acting on a (polynomially enlarged) qubit system on a 2D lattice. The only free parameters of the quantum Hamiltonian are coupling strengths of two-body interactions, which allow one to choose the type and dimension of the classical model as well as the interaction strength and temperature. This opens the possibility to study and simulate classical spin models in arbitrary dimension using a 2D quantum system.

Figure 1

(a) Illustration of mapping for a classical spin system on a triangular lattice with nearest neighbor 2-body and 3-body plaquette interactions. Quantum particles are associated with vertices (green, large), nearest neighbor pairwise interactions (red, medium) and 3-body plaquette interactions (blue, small). The classical thermal state is generated at the vertex particles (green, large). (b) 2D Cluster state with groups $A$, $B$, $C$. Deformations on group $A$ allow one to generate the state $|\phi ⟩$ on system $B\cup C$ [Fig. 1a], while deformations on group $B$ and $C$ allow one to chose interaction strengths and temperature.