# Estimating partition function using Montecarlo methods

While working on a completely unrelated quantum computing problem, I ran into a quantity that can be mapped to a partition function of spins on a triangular lattice. It is not quite an Ising model, though, since interaction happens between triplets of spins, some configurations are not allowed, and I don't explicitly have the Hamiltonian but only the weight of every configuration (I guess one could just pick a random value for the temperature and define a Hamiltonian, but it would probably just be a more complicated mess).

I have been trying to understand if I can use Montecarlo methods to get the partition function, but I am getting lost in the literature. From what I understood, my best shot would be to use something like the Wang-Landau algorithm to estimate the density of states and use that to get the partition function, but I couldn't find any paper detailing such a procedure so if someone could give me some good literature to look at (or any suggestions) that would be very helpful.

Thanks!

P.S. As far as I understand, Wang-Landau gives the density of states up to a normalization constant. However, in my specific case I have some conditions on this fictitious "partition function" that should allow me to infer that constant.

• Hi, you might want to look at my answer here to at least get started: physics.stackexchange.com/questions/550504/… Jul 23 '20 at 16:38
• What information do you have that lets you infer the normalization constant for the density of states, but doesn't let you infer the partition "function" (really just a single number, it sounds like) directly? Jul 29 '20 at 21:38
• @Daniel I have dependence on a continuous parameter $\epsilon \in [0,1]$ and I know that the "partition function" is 1 in the limit of $\epsilon \rightarrow 0$. Since the density of states is an integer in this case, then this should allow me to get the normalization constant. Jul 31 '20 at 18:18
• Ok! For $\epsilon$ near 0, you can use Monte Carlo to sample from the $\epsilon = 0$ distribution, and these samples will allow you to estimate the small-$\epsilon$ partition function pretty efficiently. Is an answer at small $\epsilon$ good enough for you? Jul 31 '20 at 19:00